Do I have to use different applications, like patches etc. for Fallout games apart of the Fallout trilogy? I just tried installing the latest patch to FOT, even extracted it right into its respective folder, yet got an error saying FOT wasn't detected on my computer.

The games that are apart of the Trilogy pack are the same as any previous versions. The difference being that some may already be patched. You should startup your games and check the version numbers in the bottom right corner to find out if your version is already upto date or not. As for FOT no longer running. Try removing the patch that caused it to break, and check if FOT works again. Maybe take a look at the readme? I'm sure you probably have, just covering all bases here.