I'm trying to restore the iPod now - presumably the same problem affecting the pausing / skipping kept interfering with the restore. Putting the iPod into disk mode shut it up enough to allow the restore. I'll post this evening to confirm whether this seems to have resolved the issue for me.
I realized as I was resetting my options that the first sign of a problem was that the iPod started speaking the names of tracks at the beginning of each track without my pressing any buttons. The pausing came later, and then the fast forwarding. It seems the issue is that the iPod is thinking buttons are being pressed (and not just one in particular). Putting the hold button on did nothing to stop this behavior, so presumably it's on the software end, not the hardware end.