An update to this problem: Firefox popped up the "welcome back" thing, and offered to "clean up" my install. Since I had recently done a backup, I decided to see what it did. It finished successfully, and Firefox has been behaving itself since then. I have NO IDEA what it did, but the problem seems to be solved, for now.
Basically it gives you a new profile. Mozilla profiles seem to easily get corrupt and cause various problems, which is why the first suggestion when tracking down a bug is to try a new profile.
Just earlier today I was thinking that what Mozilla should really be working on is an intelligent profile cleaner.
Our port is actually worse then most as we no longer seem to do some cleanup and finalizing on shutdown, which leads to more chance of profile corruption.