Well... this isn't going to be the answer you want. To be fair, there are two things I did that might have had an effect, but I really think it's just the one.
Previously, I had compressed my hard drive to save space. I uncompressed it, and uninstalled FF7 and 7th Heaven again. I restarted, then re-installed FF7 and installed 7th Heaven VERSION 2.0.
This time, it worked. I'm no programmer, so I really don't know how or why there would be a difference for some people, but it appears that the 2.2.1 version of 7th Heaven is the culprit that was preventing me from playing the game previously.