Thanks for all the help. I tried using CCleaner but still no luck. Part of the problem is the version of Windows I'm using was one I got from a Korean friend, and it's hacked to all hell. It's very difficult to reinstall things like DirectX, and even when I use an installer from another computer, it still gets an error.
What I wound up doing was just getting everything to work on an older PC with a not so great monitor, but it's better than nothing. The sad thing is, I don't even know how much I want to play this game. I really just wanted to see if I could get the characters running at a high res because it bothered me for years! I think I might give the game another shot though. I just remember its story being really convoluted, and kind of incredibly coincidental at times.
It's a shame about the music too. The orchestral version of the FF8 Battle theme is awesome, but the PC version and PSX versions sound so bland. The computers I'm using don't have any Sound Blaster cards or anything though, so I won't bother changing the music.