I can't install the newest version of quicktime because there's apparently an older one(non-functioning) that refuses to be removed.

Is there some sort of program that will completely eradicate this program from my computer? Or is there some way I can get my browser(FireFox) to use an alternate program rather than QuickTime?

What do u use Quicktime for? I hate it. I prefer using the K-Lite Mega Codec Pack instead, with its built-in Media Player Classic.

You could try QuickTime Alternative.

If you have iTunes on your PC, that includes a crappy implementation of Quicktime.
Uninstall iTunes and QT first, then install Apples full QT or the above Alternative, then re-install iTunes (without QT if it lets you).

I try not to, but a lot of websites use QuickTime to play their embedded media.

I'll look into it, will it replace QuickTime as the default embedded player?

try using CCleaner to see if there's any rouge registry entries relating to your old quicktime..