I found the answer to my door problem, but it lead to a whole new problem. It wasn't that the doors had something wrong with them. In fact, in the actual map, they're not doors. They're solid brushes, and they date from the time before I turned them into doors.
I've tried moving one of the doors. Still nothing. Hammer seems to be refusing any changes that were made since I started creating the inside of the building. And to confuse matters, every time I run the map it's playing one of my recently played songs in WMP in the background, and it did that by itself. Any ideas? Should I place this in the Problem Map vault?