I don't understand going to Derry and then backtracking to the Antrim Coast. I suggest staying near Giant's Causeway at either Bushmills, Port Ballintrae, or Port Rush on June 5th. I can heartily recommend the Bayview Hotel in Port Ballintrae as a very good hotel with an amazing view and location. It costs a bit more, but is well worth every penny. Make sure you get a room with a view.
They also have a 3 bedroom holiday condo right next door that might suit your family well. We had a bit more time than you so we walked to the Giant's Causeway, Castle Dunluce, Port Rush, and Bushmills from this location. You might want to drive to save time, but it is all just a few moments away by car.
Belfast. What are your goals there? If it the Belfast Titanic, I would stay at the Premeir Inn in the Titanic quarter. Just a few block walk and quite convenient. Lovely view out over the river and a nice walk to downtown.
I stayed at the Premeir Inn at Swords (Dublin Airport) the night before my flight out and found that it was very convenient and economical. They have shuttle that goes for 2 euros or, with a party of your size, a taxi for 8 euros might be a better choice. 5-10 minutes from the airport.
