Once admitted into short term rehab, does my father or family have any say in when he can be released?

Your father is in rehab and quite honestly this is often the ideal time to explore options. Ask to speak with "case manager" or "social worker" and please tell them what you have told us. If you need input as to whether you should seek placement for your Dad you will get it now, as to knowing what ADLs he can do ---Activities of Daily Living. They will know how much care he needs and the likelihood of a spouse being able to do it at home. They also have access to different placement suggestions. Rehabs, like hospitals, all vary in excellence. I hope you have run into a really good one with solid help and advice.

He can leave whenever he wants but please try to encourage him to stay and participate in physical therapy for his shoulder. This is important as he needs to exercise that joint and recovering from shoulder surgery is not easy for elderly folk.
His outcome at home will be greatly influenced by his work in rehab.

No you do not. You can move him anywhere you want, rehab is not jail.

Be sure and get it lined up before you make any moves, it makes it much easier if the receiving facility helps with the move.