I rented to this single guy 4 months ago. Now he tells me his college age daughter is moving in permanently. Should she be on the lease and subject to background check?
If she is over 18, then she should have to pass a basic background screening; a criminal check at the very least. If the dad is solely responsible for the rent payment and if you don't want to redo the lease completely, she could be added as an authorized occupant. If you are worried he will default on rent somehow and you want leverage, then add her onto the lease and she becomes just as responsible as he does. He really should have asked you about this ahead of time before just moving her in.
he says she is 18 and his dependent
You should figure the max occupancy for your specific situation (bedrooms, size) and place it on the lease agreements. That way there is always a max amount of people and there are no exceptions to it.