Okay, I got it figured out, it was a combination of problems.
My custom songs were in "C:\Program Files\Square Soft, Inc\Final Fantasy VII\Ficedula\custom", and FF7Music was throwing a fit over that. By moving them to F:\ff7m\custom I was able to get FF7 working again. The second thing I did was to change FF7Config.exe to run in Windows 98 compatibility mode, otherwise FF7.exe was ignoring changes made to the config, such as setting my midi to LoopBe (it was still trying to use Microsoft GS). Additionally, FF7Music was being picky about what in_mp3.dll I could use; It threw a fit every time I used the one that came with the latest version of Winamp, and simply didn't do anything with the one that came with Winamp 2.00. Eventually I got it to work using the in_mp3.dll that is bundled with Winamp 2.60.
Once I got all that taken care of, I was able to get rid of all the data in midi.lgp and still have music play. I'll see if my original problem of areas with no music disabling sound effects and FMV sound still persists at this point.
edit: It seems to be working.