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What is the purpose of th HTTP GET method in forms?

As far as I understand, the GET method asks the server to send something to the client's browser. I set up a HTTPListener in C# and when I access http://localhost:1330/form.html the request I get from the client is: GET /form.html which means that the client is saying "Hey server, I need the HTML code to display that page in the browser", which makes sense.
If I set a <form> with method=POST in form.html, the input fields values are located in the request body which is in context.Request.InputStream in C# which looks similar to this: input_name1=value&input_name2=value2&input_name3=value3... and the URL remains /form.html.
This also makes sense. The client says: "Hey server, take this data that was written in the HTML <input> elements" and the server uses it, maybe storing it in a database or computing something and send it back to the client.
Now if I set the form method to GET, the URL is modified to: /form.html?input_name1=value&input_name2=value2&input_name3=value3 and the context.Request.InputStream remains blank which is the opposite of the POST, in which the InputStream contained the data and the URL had no queries. For me, the GET method in forms doesn't make any sense. Why do we need to get the data from the form client side, send it to the server and then getting it back to client unmodified? Why do I send the data from the browser to C# and then sending it back to browser, if I can just get it client side using simple JavaScript?
In the moment the browser makes the GET request with the queries to the server, the client browser already has that data, so why does it ask the server to give it if it is already at the client's browser?

Generally speaking, an HTTP GET method is used to receive data from the server, while an HTTP POST is used to modify data or add data to a resource.
For example, think about a search form. There may be some fields on the form used to filter the results, such as SearchTerm, Start/EndDate, Category, Location, IsActive, etc, etc. You're requesting the results from the server, but not modifying any of the data. Those fields will be added to the GET request by the client so the server can filter and return the results you requested.
From the MDN article Sending form data:
Each time you want to reach a resource on the Web, the browser sends a
request to a URL. An HTTP request consists of two parts: a header that
contains a set of global metadata about the browser's capabilities,
and a body that can contain information necessary for the server to
process the specific request.
GET requests do not have a request body, so the parameters are added to the URL (this is defined in the HTTP spec, if you're interested).
The GET method is the method used by the browser to ask the server to
send back a given resource: "Hey server, I want to get this resource."
In this case, the browser sends an empty body. Because the body is
empty, if a form is sent using this method the data sent to the server
is appended to the URL.
An HTTP POST method uses the request body to add the parameters. Typically in a POST you will be adding a resource, or modifying an existing resource.
The POST method is a little different. It's the method the browser
uses to talk to the server when asking for a response that takes into
account the data provided in the body of the HTTP request: "Hey
server, take a look at this data and send me back an appropriate
result." If a form is sent using this method, the data is appended to
the body of the HTTP request.
There are plenty of resources online to learn about the HTTP protocol and HTTP verbs/methods. The MDN articles An overview of HTTP, Sending form data, and HTTP request methods should provide some good introductory reading material.

How to Submit String with 536000 Characters to API

I interpret a G code file (CNC language), serialize it into a class, and try to send in the http protocol to my API, which has a GET method.
However it is too long a string to be sent by Http.
Is there any solution to this problem? Something like compression?
Request URL Too Long
HTTP Error 414. The request URL is too long.
Using Asp.Net WebAPI

Try using POST on both sides. Doing so also makes more sense from a REST point of view, as you are submitting data, not doing a query.
Note that GET requests are usually very limited in size, see e.g. here: maximum length of HTTP GET request?

Is it possible to access parameters sent via multipart/form-data without waiting for the entire form to arrive

I have a pretty big video file I upload to a web service via multipart/form-data.
It takes ~ 30 seconds to arrive and I would prefer not waiting that long simply to access parameters I send along with the file.
My question is simple, can I access parameters sent with the form without waiting for the video payload to be uploaded?
Can this be done using headers or any other methods? 

Streaming vs. Buffering
It's about how the webserver is set up. For IIS you can enable Streaming.
Otherwise, by default, IIS will use 'buffering' - the whole request is loaded into memory first (IIS's memory that you can't get to) before your app running in IIS can get it.
Not using IIS? You have to figure out how to get the webserver to do the same thing.
How to stream using IIS:
Streaming large file uploads to ASP.NET MVC
Note the way the file is read in the inner loop:
while ((cbRead = clientRequest.InputStream.Read(rgbBody, 0, rgbBody.Length)) > 0)
{
fileStream.Write(rgbBody, 0, cbRead);
}
Here instead of just saving the data like that question does, you will have to parse any xml/json/etc or whatever contains the file parameters you speak of ... and expect the video to be sent afterwards. You can process them right away if it's a quick process ... then get the rest of the video ... or you can send them to a background thread.
You probably won't be able to parse it just dumping what you have to a json or xml parser, there will be an unclosed tag or } at the top that isn't closed til after the video data is uploaded (however that is done). Or if it's multipart data from a form submission, as you imply, you will have to parse that partial upload yourself, instead of just asking IIS for the post data.
So this will be tricky, you can first start by writing 1k at a time to a log file with a time stamp to prove that you're getting the data as it comes. after that it's just a coding headache.
Getting this to work also means you'll have to have some control over the client and how it sends the data.
That's because you'll at least have to ensure it sends the file parameters FIRST!
Which concerns me, because, if you have control of the client, why can't you take the simple route (as Nobody and Nkosi imply) and use 2 requests? You mention you need one. Why not write js client code to send the parameters first in an XHR and then the file in a second request, using a correlation ID in both to tie them together? (the server could return this from the first request and you could send it in the 2nd).
Obviously, if you're just having a form with some inputs and a file upload and doing submit, then you need one request ;-) But if you have control over the client side you're not stuck with that.
Good luck, there is some advanced programming here, but nothing super high-tech. You will make it work!!!
If you don't have control over the server code, you are probably stuck, if the server app's webserver is buffering, the server app won't get anything, of course, if you wanted to do something with the file parameters first, this really implies you have control of the server side ;-)

Pass data to a URL on a different web server without the QueryString

I've got an .ashx handler which, upon finishing processing will redirect to a success or error page, based on how the processing went. The handler is in my site, but the success or error pages might not be (this is something the user can configure).
Is there any way that I can pass the error details to the error page without putting it in the query string?
I've tried:
Adding a custom header that contains the error details, but since I'm using a Response.Redirect, the headers get cleared
Using Server.Transfer, instead of Response.Redirect, but this will not work for URLs not in my site
I know that I can pass data in the query string, but in some cases the data I need to pass might be too long for the query string. Do I have any other options?

Essentially, no. The only way to pass additional data in a GET request (i.e. a redirect) is to pass it in the query string.
The important thing to realise is that this is not a limitation of WebForms, this is just how HTTP works. If you're redirecting to another page that's outside of your site (and thus don't have the option of cookies/session data), you're going to have to send information directly in the request and that means using a query string.
Things like Server.Transfer and Response.Redirect are just abstractions over a simple HTTP request; no framework feature can defy how HTTP actually works.
You do, of course, have all kinds of options as to what you pass in the query string, but you're going to have to pass something. If you really want to shorten the URL, maybe you can pass an error code and expose an API that will let the receiving page fetch further information:
Store transaction information (or detailed error messages) in a database with an ID.
Pass the ID in the query string.
Expose a web method or similar API to allow the receiving page to request additional information.
There are plenty of hacky ways you could create the illusion of passing data in a redirect outside of a form post (such as returning a page containing a form and Javascript to immediately do a cross-domain form post) but the query string is the proper way of passing data in a GET request, so why try to hack around it?

If you must perform a redirect, you will need to pass some kind of information in the Query String, because that's how browser redirects work. You can be creative about how you pass it, though.
You could pass an error code, and have the consuming system know what various error codes mean.
You could pass a token, and have the consuming system know how to ask your system about the error information for the given token behind-the-scenes.
Also, if you have any flexibility around whether it's actually performing a redirect, you could use an AJAX request in the first place, and send back some kind of JSON object that the browser's javascript could interpret and send via a POST parameter or something like that.

A redirect is executed by most browsers as a GET, which means you'd have to put the data in the query string.
One trick (posted in two other answers) to do a "redirect" as a POST is to turn the response into a form that POSTs itself to the target site:
Response.Clear();
StringBuilder sb = new StringBuilder();
sb.Append("<html>");
sb.AppendFormat(#"<body onload='document.forms[""form""].submit()'>");
sb.AppendFormat("<form name='form' action='{0}' method='post'>",postbackUrl);
<!-- POST values go here -->
sb.AppendFormat("<input type='hidden' name='id' value='{0}'>", id);
sb.Append("</form>");
sb.Append("</body>");
sb.Append("</html>");
Response.Write(sb.ToString());
Response.End();
But I would read the comments on both to understand the limitations.

Basically there are two usual HTTP ways to send some data - GET and POST.
When you redirect to another URL with additional parameters, you make the client browser to send the GET request to the target server. Technically, your server responds to the browser with specific HTTP error code 307 + the URL to go (including the GET parameters).
Alternatively, you may want/need to make a POST request to the target URL. In that case you should respond with a simple HTML form, which consists of several hidden fields pre-filled with certain values. The form's action should point the target URL, method should be "POST", and of course your HTML should include javascript, which automatically submits the form once the document is loaded. This way the client browser would send the POST request instead of the GET one.

How to get the webpage source code using C#

I know about the WebRequest and the WebResponse objects. The problem is that I do not really want to get the source code of the webpage, I only want to check to see if the link exists or not. The thing is, if I use the GetResponse method, it goes an pull the entire source code of the site.
I am creating a broken link checker with many links. It takes quite a while to check them all. If there a way to to get MINIMAL information from a weblink? Only enough information to see if the link is valid or broken (not the entire source code).
An answer (BESIDES USING ASYNCHRONOUS TRANSFER) would be greatly appreciated!

WebRequest request = HttpWebRequest.Create("http://www.foo.com/");
request.Method = "HEAD"; // Just get the document headers, not the data.
HEAD is similar to GET, only that instead of getting the file contents, we get just the headers.

A standard way of checking the existence of a link is to use a HEAD request, which causes the remote server to send the headers for the requested object, but not the object itself. If you thus requested an object that is not on the server, the server gives you the normal 404 response, but if it does exist, you get a 200 response and no data after the headers. This way very little uninteresting data goes over the wire.