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How do you "Build" an exe that includes all the resource files like pics? Can exe be put on a web page?

I have students in high school who have created some programs using Visual Studio C#. They created some games and would like to now upload them to the web. I am pretty new to Visual Studio C#. I thought after a program was "built" that you could go into the bin/debug folder and get the exe file for the program and be able to play the game without having to have Visual Studio on the actual computer you are playing it on. It works with some of their games but some of them, there are graphical files that are missing in the exe file if all the other files aren't stored in the same place. How can I get a clean exe of their game? Can that exe be loaded up onto a web server so they can play it from there or at least download it from there?

Microsoft wrote a guide on how to do exactly this. It's on their MSDN website, which is sort of like the developer back-bone for a lot of Microsoft software and documentation.
Old guide: Adding and Editing Resources (Visual C#)
Newer guide: How to: Add or Remove Resources
The gist is that the program needs to know where the files are, relative to the location of the compiled binary (in this case, an .EXE); There's several different ways to do this, depending on your level of expertise. I would suggest that you take a look at the guides above to start that journey.

there's a few ways of doing this. (Also, make sure you're creating a 'release' build when you compile).
You can include resources in your game by creating a resources file. This is something I usually do only on winforms applications etc.
If you have on-disk files you need to distribute those on-disk files along side your game. You could do this by zipping up your game.exe and the /files/images.img folder (or what ever your resources content folder is named).
If you're feeling adventurous you could create a 'deployment project' which is a project that allows you to create an installer file. This is a bit more work however you will have fine grained control over what files go where etc.
Good luck!

EXE files generally aren't self contained for video games with many resources, just add resources in an external folder and make sure the paths to the resources in their games are not absolute but relative in the local directory.
So for example:
get rid of paths like C:/Users/Bob/Desktop/Game/Images/player.png and replace with /Images/Player.png.
Also, an EXE generally doesn't run client side on webpages, it is possible but difficult. Things such as Flash are made for this sort of issue. I'd say make the webpage a place where they can download their games. Github has a nice way to do free websites called gh-pages. OFC, you could host one yourself but that would require a lot of setup work.
Like Monza said, you can zip up the files for download. Or, you could create an installer if you wanted to be really ambitious.

I thought after a program was "built" that you could go into the bin/debug folder and get the exe file for the program and be able to play the game without having to have Visual Studio on the actual computer you are playing it on.
That is correct, given that the other machine has the necessary .NET framework version installed AND any other resources like .dll files and config files are also present. When the application is ready for release, you can set the Build options in the project properties to Release, and then when you build all the files needed for distribution will be in the bin\Release folder.
It works with some of their games but some of them, there are graphical files that are missing in the exe file if all the other files aren't stored in the same place.
This may depend on how the application was written. If resources like pictures are embedded in the assembly then those files will not be needed to run on another machine. If the application is using hard coded paths for the image resources in the source code, then the application will likely break if the necessary files aren't present when the .exe is executed.
Can that exe be loaded up onto a web server so they can play it from there or at least download it from there?
Yes - you could upload the .exe to a file server to make available for a download, but you would not be able to run it within a browser over the web. I would recommend zipping up the .exe and other files needed and hosting the .zip file on the web server for download; browsers may give a warning or block downloading an .exe .
Hope this explains it a little bit for you.

Creating a "portable" .exe (without installer)

I've recently coded a little program to determine numbers in a picture and it is reliant on two libraries I've used. (DLLs)
Since my target computer is not allowed to install programs due to security reasons, I need to create a portable .exe.
.NET is installed on the target computer but for some reason VS still does not include the libraries I've used in the exe but instead creates an application folder with a setup.exe, some .DEPLOY files and an application manifest.
I am new to VS and .NET in general so this question could be easy to answer, but I'm asking since I've found nothing useful on StackOverflow neither on google.

You can simply build the application and copy your bin/Debug folder along, but that would still mean you need multiple files.
In order to merge all references into the executable, use ILMerge. Here is some help calling ILMerge.
Basically, after building, you should do something like this:
ilmerge /target:winexe /out:SelfContainedProgram.exe
Program.exe ClassLibrary1.dll ClassLibrary2.dll
There is just one file you need to send along.

One way to do this is to build your application in Release mode (You can pick from Debug or Release in the drop-down). Then go to C:\Projects\[ProjectName]\[ProjectName]\bin\Release (The location of your project folder may vary). You'll see a bunch of files but all you really need are the DLLs, executable, and the config if you used one. You won't have to do any setup if you keep the necessary files in the application's folder, just copy them all to a folder on the target computer, create a shortcut if you want then you're good to go.

You can just copy all your assemblies into any folder you want. Simply chose "Build" from within Visual Studio and copy the files from bin/debug to your destination-folder.
However you have to ensure that all (relative) paths (if existing) still work as you cannot be sure where the user of your program copies the files to.

One simple way could be to use 7zip Packager, it doesn't need any installer. However, VisualStudio method might be more reliable.

I encountered the same issue recently. ILMerge suggestion above is no longer supported. I found Fody.Costura as a modern replacement.

Bundle several cabinet files together? (for Windows Mobile)

I am writing application for Windows Mobile 6.5, and for application to run, I have to install sqlce.wce5.armv4i.cab and NETCFv35.wm.armv4i.cab files on the end device.
Is there someway that I can combine and bundle those two cabinet files together with my application cabinet file? I want to create a single install process, which would install each file sequentially one after another (when I start bundled cabinet file)?
Is it possible? Or what is the easier solution to create more automated installation of required packages (such as for SQL Server CE)?
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I am able to create cab file with other two cab files in it, but how can I make them execute right after I finish installing cab file that contains them? Right now it just extract them.

If I am right you are looking for Multicab Installation.
WM already comes with an example on this. Here is a reference.
The sample can be found on your PC at C:\Program Files\Windows Mobile 6.5.3 DTK\Samples\Common\CPP\Win32\multicabinstall

From what I have learned in the past, that can't really be done.
When I learned this long ago (back on some other site before SO came along), I was bummed and thought this was the dumbest thing ever.
Now, I keep those two files (plus another I use called NETCFv35.Messages.EN.wm.cab) on the same Network folder that my installing CAB file is located on.
Also, I install those files onto each device's Flash Drive, that way if one of our employees forgets to charge the battery and the device resets, I can quickly get the old WM5 device's CF upgraded to v35 and and get sqlce running.
Note:
Things may have changed! Someone may have learned a way to combine these CAB files into your project's cab file.
If you can do that, just keep in mind that every time you go to update your application, instead of your CAB file being 200k, it is going to have to be the combined size of all of the CABs.
You might run out of room!
Also, if one of your CAB files is going to be the NETCFv35.wm.armv4i.cab (at 2,644 KB), the device's OS has to reboot after installation. That would likely cause your program's CAB file to fail.
There may be a way around that, but not that I've learned.

Updater.exe built into app.exe

Currently I have 2 exe files. app.exe and updater.exe. When app.exe finds out that there is new version available it runs updater.exe, which downloads and replaces it.
I'm wondering if it's possible to build updater.exe into app.exe. On app start it should check if there is updater.exe in the directory and if it's not than extract it. Any help appreciated.

You can integrate updater.exe as a resource into app.exe. The following SO post shows how to extract it at run-time:
Embedding an external executable inside a C# program
Note that, if your application is located in the default application directory (C:\Program Files, or, more generally, %ProgramFiles%), you will not have permissions to create a file in the same directory (which is a good thing). Thus, you might need to extract updater.exe into a temporary directory that the current user has write permissions to (such as Path.GetTempPath).

When I'm understanding you correctly you need to pack you updater.exe into you app.exe as ressource. Extract it, when App.exe starts.
But I can tell you, this idea with replace is bad. You should choose a MSI package to deploy your application.

For traversing and listing directory contents, see http://msdn.microsoft.com/en-us/library/system.io.directory.aspx
For accessing your embedded resources, see http://msdn.microsoft.com/en-us/library/xc4235zt.aspx
For writing binary data to disk, see http://msdn.microsoft.com/en-us/library/system.io.binarywriter.aspx
For running an exe from disk, see http://msdn.microsoft.com/en-us/library/system.diagnostics.process.aspx

What is the best way for on demand download and install of dlls in .NET?

We have an application that we wish to install just a basic shell for. As the users use it, it will download and install the necessary dlls for actions that need to take place (imagine a wizard application scenario with several possible paths). Currently, everything is installed for all possible paths through the shell app.
For about 3 months out of the year, the dlls used for the possible paths go through a high churn rate with updates, so we wish to start ensuring the users have the most recent version of these dlls. The idea is that after all their selections are made, we'd make a web check to see what dlls are required for their selection and check to make sure they have the most recent version of those files.
All of this we have a plan for on how to do it. The problem I'm fighting is what is the most appropriate way to "install" these files. ClickOnce is not an option...too much legacy stuff here. Our app is installed in "Program Files" which obviously has restrictions for writing random files into the program's install folder under Vista and later.
Right now I see the options as the following:
On install mark the install directory as writable for the "Everyone" group. I haven't actually tested to see if this would work yet, or if Vista does something different in this scenario.
Split the download portion out into a second app that we can have prompt for elevated privileges so that it can download and install these files.
I'm leaning towards the second option since that maintains the security aspect of the Program Files folder. Others in the group lean towards the first option because they just don't want to have to worry about things. Or is there some other option I'm missing?
The app is a .NET app, though it has some requirements of third party dlls that are not managed assemblies.

As long as the dlls you want to load are managed, there are several ways to do this.
One way is to designate an Environment.SpecialFolder path such as AppData and dynamically load assemblies into your appdomain from there. RssBandit does this for plugins, there is a special directory that loads dlls and searches for specific interface implementations, loads them into a temporary appdomain, and then calls them from the app. You could take this one step further by using an IoC library like ninject or structuremap.
you could also try MEF, the new extensibility framework coming out in C# 4.

Personally, I like what Firefox/xul apps do. It's a hybrid of your two proposed solutions, I suppose. They have an updater.exe that lives in the install directory. I assume that means that the install directory is made writable during install so that they can run the update application. However, having never deployed a application in this manner, I can't tell you how much of a headache (or not) it is.
An alternative which you haven't mentioned, and may not know about, is using the .Net download cache. When you attempt to load an assembly you can give it a code base to load from. If you set the codebase to a web url (i.e. http://mywebhost/mycoolapp/) .Net will download the assembly from that url if it's not found in the download cache. It will also grab the latest version of the assembly from the web url if there is one.
This approach can be a pain as you'll likely have to deal with CAS security issues if your app needs elevated permissions. However, it is nice not having to write code to download the latest versions of your assemblies for you. If you want more information, I can find some resources and give more detailed examples.

The way I handle it is to have an update.exe installed into program files next to the main .exe file.
Then, on app startup, I have the app download an xml file off of the web and save it in the App Data folder. This file contains the latest versions of the dlls and has a simple Filename, Version structure.
Run through the list of filenames and if you don't have the dll locally or you have an older version, then add the needed dll to an update list.
After you generate your update list. Fire off the Updater.exe with a command line list of the files to be updated. You don't have to write them to program files but I do. On Vista, my updater pops up the UAC prompt correctly (as it should to maintain Program Files' security).
The updater then downloads the files to Program Files and restarts the main app.
One problem with firing off the second app is that you have to give it a manifest with "AsAdministrator" set in it.
This isn't hard to do, but once the updater is done and retriggers the main app, it cannot start the main app with normal privileges. An exe running as administrator can only start other exe's as administrator also, even if "AsInvoker" is set in the manifest. I don't know why you can't restrict it back to normal rights...you can only elevate permissions for some reason...