Am I correct to assume that I can delete the older copies? (I read somewhere that I should never delete any wallet.)
As long as you still have backups that are more recent versions of the backup that you are deleting, yes, you can delete those. However you should still keep several recent versions of the backup in multiple places to be safe.
Why would the different wallet.dat files show different balances? Don't they have the same addresses and keys?
No. After 100 addresses are used (including change addresses), a new set of 100 addresses will have been generated which are not part of the previous backup.