Painting the primer first isn't necessary. I do recommend 2 coats of paint on the entire house. 2 coats will last longer and the final look will be more consistent.
It is best if the primer was applied before the caulking. Caulking sticks well to most surfaces but sticks even better to primer. The only caulking that requires priming is some 1 part urethane's, never use 100% silicone it isn't paintable.