Will I have completely normal use of my hand after shoulder surgery?

You should. If you have normal use of your hand prior to surgery there is no reason you won't have normal function of your hand post surgery. Depending on the surgery you may have some restrictions of motion or weight of that arm, but you should be able to use hand for normal activities.
Yes. As long as you had normal use prior to surgery. The only way the hand function might be altered is if you had any nerve work or nerve injury during surgery, otherwise the hand should be fine.