How exactly do RAM test applications work, and is it possible to write such using C# (Example)?
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You basically write to the RAM, read it back and compare this with the expected result. You might want to test various patterns to detect different errors (always-0, always-1), and run multiple iterations to detect spurious errors.
You can do this in any language you like, as long as you have direct access to the memory you want to test. If you want to test physical RAM, you could use P-invoke to reach out of the CLR.
However, this won’t solve one specific problem if your computer is based on the Von Neumann architecture: The program that tests the memory is actually located inside the very same memory. You would have to relocate the program to test all of it. The German magazine c’t found a way around this issue for their Ramtest: They run the test from video memory. In practice, this is impossible with C#.