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How does LINQ's OrderBy jive with MoveNext?

This thread says that LINQ's OrderBy uses Quicksort. I'm struggling how that makes sense given that OrderBy returns an IEnumerable.
Let's take the following piece of code for example.
int[] arr = new int[] { 1, -1, 0, 60, -1032, 9, 1 };
var ordered = arr.OrderBy(i => i);
foreach(int i in ordered)
Console.WriteLine(i);
The loop is the equivalent of
var mover = ordered.GetEnumerator();
while(mover.MoveNext())
Console.WriteLine(mover.Current);
The MoveNext() returns the next smallest element. The way that LINQ works, unless you "cash out" of the query by use ToList() or similar, there are not supposed to be any intermediate lists created, so each time you call MoveNext() the IEnumerator finds the next smallest element. That doesn't make sense because during the execution of Quicksort there is no concept of a current smallest and next smallest element.
Where is the flaw in my thinking here?

the way that LINQ works, unless you "cash out" of the query by use ToList() or similar, there are not supposed to be any intermediate lists created
This statement is false. The flaw in your thinking is that you believe a false statement.
The LINQ to Objects implementation is smart about deferring work when possible at a reasonable cost. As you correctly note, it is not possible in the case of sorting. OrderBy produces as its result an object which, when MoveNext is called, enumerates the entire source sequence, generates the sorted list in memory and then enumerates the sorted list.
Similarly, joining and grouping also must enumerate the whole sequence before the first element is enumerated. (Logically, a join is just a cross product with a filter, and the work could be spread out over each MoveNext() but that would be inefficient; for practicality, a lookup table is built. It is educational to work out the asymptotic space vs time tradeoff; give it a shot.)
The source code is available; I encourage you to read it if you have questions about the implementation. Or check out Jon's "edulinq" series.

There's a great answer already, but to add a few things:
Enumerating the results of OrderBy() obviously can't yield an element until it has processed all elements because not until it has seen the last input element can it know that that last element seen isn't the first it must yield. It also must work on sources that can't be repeated or which will give different results each time. As such even if some sort of zeal meant the developers wanted to find the nth element anew each cycle, buffering is a logical requirement.
The quicksort is lazy in two regards though. One is that rather than sort the elements to return based on the keys from the delegate passed to the method, it sorts a mapping:
Buffer all the elements.
Get the keys. Note that this means the delegate is run only once per element. Among other things it means that non-pure keyselectors won't cause problems.
Get a map of numbers from 0 to n.
Sort the map.
Enumerate through the map, yielding the associated element each time.
So there is a sort of laziness in the final sorting of elements. This is significant in cases where moving elements is expensive (large value types).
There is of course also laziness in that none of the above is done until after the first attempt to enumerate, so until you call MoveNext() the first time, it won't have happened.
In .NET Core there is further laziness building on that, depending on what you then do with the results of OrderBy. Since OrderBy contains information about how to sort rather than the sorted buffer, the class returned by OrderBy can do something else with that information other than quicksorting:
The most obvious is ThenBy which all implementations do. When you call ThenBy or ThenByDescending you get a new similar class with different information about how to sort, and the sort the OrderBy result could have done probably never will.
First() and Last() don't need to sort at all. Logically source.OrderBy(del).First() is a variant of source.Min() where del contains the information to determine what defines "less than" for that Min(). Therefore if you call First() on the results of an OrderBy() that's exactly what is done. The laziness of OrderBy allows it to do this instead of quicksort. (Which means O(n) time complexity and O(1) space complexity instead of O(n log n) and O(n) respectively).
Skip() and Take() define a subsequence of a sequence which with OrderBy must conceptually happen after that sort. But since they are lazy too what can be returned is an object that knows; how to sort, how many to skip, how many to take. As such partial quicksort can be used so that the source need only be partially sorted: If a partition is outside of the range that will be returned then there's no point sorting it.
ElementAt() places more of a burden than First() or Last() but again doesn't require a full quicksort. Quickselect can be used to find just one result; if you're looking for the 3rd element and you've partitioned a set of 200 elements around the 90th element then you only need to look further in the first partition and can ignore the second partition from now on. Best-case and average-case time complexity is O(n).
The above can be combined, so e.g. .Skip(10).First() is equivalent to ElementAt(10) and can be treated as such.
All of these exceptions to getting the entire buffer and sorting it all have one thing in common: They were all implemented after identifying a way in which the correct result can be returned after making the computer do less work*. That new [] {1, 2, 3, 4}.Where(i => i % 2 == 0) will yield the 2 before it has seen the 4 (or even the 3 it won't yield comes from the same general principle. It just comes at it more easily (though there are still specialised variants of Where() results behind the scenes to provide other optimisations).
But note that Enumerable.Range(1, 10000).Where(i => i >= 10000) scans through 9999 elements to yield that first. Really it's not all that different to OrderBy's buffering; they're both bringing you the next result as quickly as they can†, and what differs is just what that means.
*And also identifying that the effort to detect and make use of the features of a particular case are worth it. E.g. many aggregate calls like Sum() can be optimised of the results of OrderBy by skipping the ordering completely. But this can generally be realised by the caller and they can just leave out the OrderBy so while adding that would make most calls to Sum() slightly slower to make that case much faster the case that benefits shouldn't really be happening anyway.
†Well, pretty much as quickly. It would be possible to get the first results back more quickly than OrderBy does—when you've got the left most part of a sequence sorted start giving out results—but that comes at a cost that would affect the later results so the trade-off isn't necessarily that doing that would be better.

It is wise to create a variable to avoid using several times a count?

Many times I'm in front of a code like that:
var maybe = 'some random linq query'
int maybeCount = maybe.Count();
List<KeyValuePair<int, Customer>> lst2scan = maybe.Take(8).ToList();
for (int k = 0; k < 8; k++)
{
if (k + 1 <= maybeCount) custlst[k].Customer = lst2scan[k].Value;
else custlst[k].Customer = new Customer();
}
Each time I have a code like this. I ask me, must I create a variable to avoid the for-each calculate the Count() ?. Maybe for only 1 Count() in the loop it's useless.
Somebody have an advice with the "right way" to code that in a loop. Do you do in case per case or ?
In this case, is my maybeCount variable useless ?
Do you know if the Count() count each time or he just return the content of a count variable.
Thanks for any advice to improve my knowledge.

If the count is guarranteed to not change then yes this will stop the need to calculate the count multiple times, this can become more important when the Count is resolved from method that can take some time to execute.
If the count does change then this can result in erroneous results being generated.
In answer to your questions.
The way you have done it looks pretty reasonable
As mentioned, your cnt variable means that you won't have to repeat your linq query on each iteration.
The count will be determined each time you call it, it has no memory for what the count is
Note: It is important to note that I am talking about the Enumerable.Count method. The List<T>.Count is a property that can just return a value

Well, it depends ;)
I wouldn't use a loop. I would do something like this:
-
var maybe = 'some random linq query'
// if necessary clear list
custlst.Clear();
custlst.AddRange(maybe.Take(8).Select(p => p.Value));
custlst.AddRange(Enumerable.Range(0, 8 - custlst.Count).Select(i => new Customer()));
-
In this case, yes.
It depends if the underlying type is a List (a Collection, in fact) or a simple enumerable. The implementation of Count will look for a precomputed Count value and use it. So, if you're cycling a list, the variable is almost useless (still a little faster, but I don't think it's relevant by any means), if you're cycling a Linq query, a variable is recommended, because Count is going to cycle the entire enumeration.
Just my 2c.
Edit: for reference, you can find the source for .Count() at https://github.com/dotnet/corefx/blob/master/src/System.Linq/src/System/Linq/Enumerable.cs (around line 1489); as you see, it checks for ICollection, and uses the precomputed Count property.

Does LINQ cache computed values?

Suppose I have the following code:
var X = XElement.Parse (#"
<ROOT>
<MUL v='2' />
<MUL v='3' />
</ROOT>
");
Enumerable.Range (1, 100)
.Select (s => X.Elements ()
.Select (t => Int32.Parse (t.Attribute ("v").Value))
.Aggregate (s, (t, u) => t * u)
)
.ToList ()
.ForEach (s => Console.WriteLine (s));
What is the .NET runtime actually doing here? Is it parsing and converting the attributes to integers each of the 100 times, or is it smart enough to figure out that it should cache the parsed values and not repeat the computation for each element in the range?
Moreover, how would I go about figuring out something like this myself?
Thanks in advance for your help.

LINQ and IEnumerable<T> is pull based. This means that the predicates and actions that are part of the LINQ statement in general are not executed until values are pulled. Furthermore the predicates and actions will execute each time values are pulled (e.g. there is no secret caching going on).
Pulling from an IEnumerable<T> is done by the foreach statement which really is syntactic sugar for getting an enumerator by calling IEnumerable<T>.GetEnumerator() and repeatedly calling IEnumerator<T>.MoveNext() to pull the values.
LINQ operators like ToList(), ToArray(), ToDictionary() and ToLookup() wraps a foreach statement so these methods will do a pull. The same can be said about operators like Aggregate(), Count() and First(). These methods have in common that they produce a single result that has to be created by executing a foreach statement.
Many LINQ operators produce a new IEnumerable<T> sequence. When an element is pulled from the resulting sequence the operator pulls one or more elements from the source sequence. The Select() operator is the most obvious example but other examples are SelectMany(), Where(), Concat(), Union(), Distinct(), Skip() and Take(). These operators don't do any caching. When then N'th element is pulled from a Select() it pulls the N´th element from the source sequence, applies the projection using the action supplied and returns it. Nothing secret going on here.
Other LINQ operators also produce new IEnumerable<T> sequences but they are implemented by actually pulling the entire source sequence, doing their job and then producing a new sequence. These methods include Reverse(), OrderBy() and GroupBy(). However, the pull done by the operator is only performed when the operator itself is pulled meaning that you still need a foreach loop "at the end" of the LINQ statement before anything is executed. You could argue that these operators use a cache because they immediately pull the entire source sequence. However, this cache is built each time the operator is iterated so it is really an implementation detail and not something that will magically detect that you are applying the same OrderBy() operation multiple times to the same sequence.
In your example the ToList() will do a pull. The action in the outer Select will execute 100 times. Each time this action is executed the Aggregate() will do another pull that will parse the XML attributes. In total your code will call Int32.Parse() 200 times.
You can improve this by pulling the attributes once instead of on each iteration:
var X = XElement.Parse (#"
<ROOT>
<MUL v='2' />
<MUL v='3' />
</ROOT>
")
.Elements ()
.Select (t => Int32.Parse (t.Attribute ("v").Value))
.ToList ();
Enumerable.Range (1, 100)
.Select (s => x.Aggregate (s, (t, u) => t * u))
.ToList ()
.ForEach (s => Console.WriteLine (s));
Now Int32.Parse() is only called 2 times. However, the cost is that a list of attribute values have to be allocated, stored and eventually garbage collected. (Not a big concern when the list contains two elements.)
Note that if you forget the first ToList() that pulls the attributes the code will still run but with the exact same performance characteristics as the original code. No space is used to store the attributes but they are parsed on each iteration.

It has been a while since I dug through this code but, IIRC, the way Select works is to simply cache the Func you supply it and run it on the source collection one at a time. So, for each element in the outer range, it will run the inner Select/Aggregate sequence as if it were the first time. There isn't any built-in caching going on -- you would have to implement that yourself in the expressions.
If you wanted to figure this out yourself, you've got three basic options:
Compile the code and use ildasm to view the IL; it's the most accurate but, especially with lambdas and closures, what you get from IL may look nothing like what you put into the C# compiler.
Use something like dotPeek to decompile System.Linq.dll into C#; again, what you get out of these kinds of tools may only approximately resemble the original source code, but at least it will be C# (and dotPeek in particular does a pretty good job, and is free.)
My personal preference - download the .NET 4.0 Reference Source and look for yourself; this is what it's for :) You have to just trust MS that the reference source matches the actual source used to produce the binaries, but I don't see any good reason to doubt them.
As pointed out by #AllonGuralnek you can set breakpoints on specific lambda expressions within a single line; put your cursor somewhere inside the body of the lambda and press F9 and it will breakpoint just the lambda. (If you do it wrong, it will highlight the entire line in the breakpoint color; if you do it right, it will just highlight the lambda.)

C# Linq Where(expression).FirstorDefault() vs .FirstOrDefault(expression)

What is the difference between these two Linq queries:
var result = ResultLists().Where( c=> c.code == "abc").FirstOrDefault();
// vs.
var result = ResultLists().FirstOrDefault( c => c.code == "abc");
Are the semantics exactly the same?
Iff sematically equal, does the predicate form of FirstOrDefault offer any theoretical or practical performance benefit over Where() plus plain FirstOrDefault()?

Either is fine.
They both run lazily - if the source list has a million items, but the tenth item matches then both will only iterate 10 items from the source.
Performance should be almost identical and any difference would be totally insignificant.

The second one. All other things being equal, the iterator in the second case can stop as soon as it finds a match, where the first one must find all that match, and then pick the first of those.

Nice discussion, all the above answers are correct.
I didn't run any performance test, whereas on the bases of my experience FirstOrDefault() sometimes faster and optimize as compare to Where().FirstOrDefault().
I recently fixed the memory overflow/performance issue ("neural-network algorithm") and fix was changing Where(x->...).FirstOrDefault() to simply FirstOrDefault(x->..).
I was ignoring the editor's recommendation to change Where(x->...).FirstOrDefault() to simply FirstOrDefault(x->..).
So I believe the correct answer to the above question is
The second option is the best approach in all cases

Where is actually a deferred execution - it means, the evaluation of an expression is delayed until its realized value is actually required. It greatly improves performance by avoiding unnecessary execution.
Where looks kind of like this, and returns a new IEnumerable
foreach (var item in enumerable)
{
if (condition)
{
yield return item;
}
}
FirstOrDefault() returns <T> and not throw any exception or return null when there is no result

finding all lines in a list that contain x or y?

can I do this without looping through the whole list?
List<string> responseLines = new List<string>();
the list is then filled with around 300 lines of text.
next I want to search the list and create a second list of all lines that either start with "abc" or contain "xyz".
I know I can do a for each but is there a better / faster way?

You could use LINQ. This is no different performance-wise to using foreach -- that's pretty much what it does behind the scenes -- but you might prefer the syntax:
var query = responseLines.Where(s => s.StartsWith("abc") || s.Contains("xyz"))
.ToList();
(If you're happy dealing with an IEnumerable<string> rather than List<string> then you can omit the final ToList call.)

var newList = (from line in responseLines
where line.StartsWith("abc") || line.Contains("xyz")
select line).ToList();

Try this:
List<string> responseLines = new List<string>();
List<string> myLines = responseLines.Where(line => line.StartsWith("abc", StringComparison.InvariantCultureIgnoreCase) || line.Contains("xyz")).ToList();
The StartsWith and Contains shortcut - the Contains will only evaluate if the StartsWith is not satisfied. This still iterates the whole list, but of course there is no way to avoid that if you want to check the whole list, but it saves you from doing typing a foreach.

Use LINQ:
List<string> list = responseLines.Where(x => x.StartsWith("abc") || x.Contains("xyz")).ToList();

Unless you need all the text for some reason, it would be quicker to inspect each line at the time when you were generating the List and discard the ones that don't match without ever adding them.
This depends on how the List is loaded as well - that code is not shown. This would be effective if you were reading from a text file since then you could just use your LINQ query to operate directly on the input data using File.ReadLines as the source instead of the final List<string>.
var query = File.ReadLines("input.txt").
Where(s => s.StartsWith("abc") || s.Contains("xyz"))
.ToList();

LINQ works well as far as offering you improved syntax for this sort of thing (See LukeH's answer for a good example), but it isn't any faster than iterating over it by hand.
If you need to do this operation often, you might want to come up with some kind of indexed data structure that watches for all "abc" or "xyz" strings as they come into the list, and can thereby use a faster algorithm for serving them up when asked, rather than iterating through the whole list.
If you don't have to do it often, it's probably a "premature optimization."

Quite simply, there is no possible algorithm that can guarantee you will never have to iterate through every item in the list. However, it is possible to improve the average number of items you need to iterate through - sorting the list before you begin your search. By doing so, the only times you would have to iterate through the entire list would be when it is filled with only "abc" and "xyz."
Assuming that it's not practical for you to have a pre-sorted list by the time you need to search through it, then the only way to improve the speed of your search would be to use a different data structure than a list - for example, a binary search tree.