The domain does not determine the OS of the desktop, but it does limit it. Windows ME, 95, and (* can not operate of a Active Directory domain (200 or 2003) that has been upgraded to Native mode from the default of mixed mode.
Well, usually the network administrators that manage computers on domains which is the only reason I would ever have a computer under a domain within a network, they can see statistics about the computer remotely, which does include the operating system they are running, and yes it is stored within the registry. There are more details about your question, but I think your best approach to understanding how and why computers are members of a domain, is to begin with the MCTS or CCNA. Best of luck, G0TR00T.