Had a trusted mechanic look at it. Here is the latest:
1. Bad motor mount.
2. (Here's where you get points) Dragging caliper.
(1) he noticed it clunked when putting it into reverse. Now I know where that sound came from.
Point to learn about (2), I guess, is that they get old enough and just don't retract as they ought. Sometimes, it's because brake dust had found its way inside the piston boot. Sometimes, it's just because...
Will keep you posted. He'll replace the mount first.