No, for dumping the controlfile to trace you don't have to shut down the database - in fact if you do you won't be able to create the trace file ;-)). For isuing ALTER DATABASE BACKUP CONTROLFILE TO TRACE the instance must know the contents of the controlfile, so the database must be open or at least in MOUNT state. There also isn't any point in backing up a database just because you are dumping the controlfile to trace (although there can never be too much of *valid* backups ;-)) ).

For other isues regarding the cloning process others have specified just about everything...