
Rothbury is nice, you'll like it. Nursey has hit the big sights adn I would take in both Cragside and Alnwick.
Cragside is great but check to see if the restoration work is finished. The grounds are lovely there and there are some nice walks around the house. There's also Wallington Hall and Lindisfarne castle (you can go drink some famous mead) If you like history you could go out the Tyne valley and see Housesteads and have a wander around Hexham or Corbridge. Alnmouth or Druridge Bay is good for taking the dogs to the beach. There's a list of things to do with some links
here.
I don't know about the pubs but you will find once you're out of the city, quite a few in northumberland let your dog in.