UGerstl and Leonhart, your posts actually solved my problem. For some reason, in the FF7Music configuration, the profile was on OST Remastered instead of Bootleg while fixing the OpenGL cfg file fixed the battle interface (at least I think it did. It looks better than before:
http://img17.imageshack.us/img17/4035/ff7battleproper.png). My problems are now fixed and FF7 is playable. Thanks a lot for your help!
I will point out though that there are some unstable points that causes the game to hang (and in my case, it destroyed the installation). When customizing the controls, I ran into one point where the game semi-froze on me. The music and everything else was fine, but my controls were non-responsive, so I used the Task Manager to end the game. When I restarted, my controls refused to work again, so I couldn't load a save file, forcing me to completely reinstall FF7 and the mods. I got around that this time by being really careful while customizing the controls after reinstalling.
Basically, what I noticed that exiting via the Task Manager will destroy the installation, although if the game hangs and your controls become unresponsive for some reason, forcing you to quit via the Task Manager, you might as well just reinstall everything because the controls refuse to work again when you restart the game.