No hidden files that I know of. They will have ".fp7" all the way up through FM11, I think version 12 finally forces the suffix to "fp12."
As for where the files go, you get a dialog asking where to put any new database as soon as you start creating it. That's the standard OS Navigation dialog. They will go where that defaults to or where you select.
Un-deleting is a risky endeavor, however, if the drive has been written to after a deletion, any or all the deleted bits may have been over-written. Even if that's just one sector, it may be impossible to find all the others. I'm assuming the deletion may have happened some time ago. In that case, Time Machine should have a copy saved from before that time, assuming your backups go back that far. Of course, any other non-destructive (one that adds new stuff without deleting anything) backups should work as well.
Good luck!