Any jewelry store should be able to replace the battery. Honestly, you should be able to do it yourself. You can buy one at the jewelry counter at Wally-World once you have it in your hand.
I doubt it. They only sell them. They don't do repairs. Depending on the watch, you could probably get a new battery at your local Walmart, or at a jewellery store near you that sells watches. If it is one of the high end watches, though, you sometimes have to go through an authorized repair site.