When I install the HDD into a different case or a PC, can I get access to the data?
No. Your data is being saved encrypted. Also you will find no valid partitions. Data recovery tools will fail, too.
Can't a third person just scan his fingerprint to get access to the data?
No. New fingerprints can only be scanned when beforehand a valid fingerprint has been scanned. The only security risk is the password, which also secures your data. You should choose a well-chosen and secure one. The included setup tool only allows to (re-)setup the HDD. If a person tries to use this tool to scan his fingerprint, the data on the HDD will be lost.
I try to set up my HDD with enclosed software, but the tool requires a fingerprint.
This is additional security. When a harddrive has already been registered (e.g. for testing), at least one fingerprint keeps saved. This prevents, that a user might get access to saved data by going a way round. You must completely delete the hard drive. repartitioning and reformatting does not work here. For this you can use Active@ Killdisk.