Correct. MySQL databases are stored on a separate server from your files. We do move these around on occasion, but the process is largely invisible.
All correct. We keep your files on the original server for at least a week or so as a safeguard against issues during the move, after which they are removed. Unless we're notified of a problem, though, we don't perform any extra copies of data after the move is complete — any changes which you made on the old server will be lost.