In the Modify User menu, there is the drop down menu for what group they belong to and some check boxes for other groups for added access. So, if I leave the user in the Registered group in the drop down menu I can still give them permissions to other group albums. The reverse doesn't seem possible since there is no check box for Registered users. Is there a way to assign a user to a different group but still give them access to the Registered albums?
No, not with the current coding of Coppermine.
The 4 default groups are meant to be base groups. If you want your user to be in one of them, select it as the "primary group" in the dropdown box. Then add secondary groups as desired. If you don't want the user to be one of them, select a different group in the dropdown box.
I'm not sure why you would need it the other way around. Isn't this equivalent?