I believe the Alamo/National counter is open at the Dolphin from 8am-4pm. But, if you get there and do the paperwork during the hours they are open, you can return later to get the car..that's what I have done.
The Car Care Center will not deliver a car to you...you have to get to them. But, if you call them prior to your reservation time, they can provide a shuttle ride for you...they will come to you, and then take you to the CCC so you can get your car.