Is Galveston back to normal?

Normal, not quite. But back, yes definitely. There are plenty of things that are open for you to do and see. Most businesses along the Seawall are open. Downtown opens more and more businesses every day. You can definitely bring your dog to the beach. Although, hold on to her, because at that size she might get swept or blown away. :) As far as attractions go, I can't think of any that aren't outdoors that will allow dogs.