Okay! I have something to blame beside gremlins, and that's Steam!
The upshot seems to be this: When I launch the game from Steam, I can't control the mouse cursor; when I launch it by opening the executable file ( C:\Program Files (x86)\Steam\SteamApps\common\Blackwell Unbound\Unbound.exe ), it works fine -- but, annoyingly, it seems that I can't get any achievements.
Anyway, hope this helps anyone who is trying to track down this problem.