Is there a way to have MC update (and drop its database listing automatically) when files move?
There are two basic ways:
1. Move the files using MC. MC has a wonderful and powerful tool called Rename, Move, and Copy files. You just select the files, run the tool, and configure it, and it will do the the moves and the database updates for you. Depending on exactly how you're doing your moves (preserving directory structures, or reorganizing), this might be a great tool for you.
http://wiki.jriver.com/index.php/Rename,_Move,_and_Copy_Files2. Auto import. If you set up Auto Import to import from both your C: location *and* your H: drive, it should sort out anything you move externally, within reason. This should be relatively seamless. You might want to try it with a few files first and see how it works.
Regarding moving the drive to another computer: I'm assuming these are totally separate installations of MC that share nothing but the drive. In that case, you shouldn't have any issues, except for any special tags that you write on the first computer, that might not be copied into your files. Something like a custom field you defined, or some other field that can't be saved into the music files. Otherwise, as long as the changes are being written to the files, you should be fine.
Moving the drive back and forth might have some implications for play counts, ratings, and that kind of thing, depending on how you use MC on each computer. If they both write to the files, the last one to write "wins".
Brian.