Make sure you are only making copies of backup databases and not live FileMaker databases or you can corrupt them.
You specify in the Server Admin console where you want the backups to go. But if you want the default locations, they are:
\Program Files (x86)\FileMaker\FileMaker Server\Data\Backups\
You can copy a database from the Backups. You cannot rename it from the Admin Console, but you can using Windows Explorer or Mac OS X Finder. But you can use the Admin Console to export one and rename it if that helps.