It sounds like a fantastic trip to me ! I have sailed on both the Dawn Princess and the Grand Princess several times.
I think that you will find the rooms comparable, although the balcony on the Grand may be larger.
There is a fantastic site that I use for ships comparison, and it says that BC on both ships are the same square footage, but in their depiction of the cabin, the balcony looks larger on the Grand.
Check out: <http://www.cruisestateroom.com/>
I love both the ships, but actually prefer the Dawn Princess, just due to size.
As far the airport, you will want the San Juan airport, and yes, at luggage claim, there will be a transportation person that will give you a receipt (they have fare controls in San Juan) with what you should pay and direct you to the cab line.
As far as after the cruise, I'd purchase the cruise line transfers.
Hopefully you will be able to go to San Juan at least a day early, it is one of my favorite places.
