does your dog have to be licensed to come to the park

Yes. He is supposed to have his tag on his collar.

Yes and all shots.

I live in Alabama now, but there used to be a sign that said, current on vaccines and a license. But it's been a few years since I've been there

Hello, I had not seen a sign that said license for rabies a requirement, but it might be a good thing to have if going to have your dog with other dogs. It does state that any signs of dog aggression are asked to exit the premises though.