Hey dwalker, pretty much all places in Asheville are kid and dog friendly! I'd really only be concerned about it if you intend to go to upper end restaurants. If there are certain types of food you're keen on, let me know and I might be able to direct you better. As for breweries, you'll be spoilt for choice.
Downtown:
Wicked Weed/The Funkatorium- Great for sours, IPAs, odd-ball beers
The Thirsty Monk- Brews their own beer and also has two floors each with their own bar. The bottom bar is totally Belgian beers and the top is best of American and local/regional breweries
Burial Brewery- Belgians and Stouts are their strengths
Hi-Wire Brewing/Twin Leaf/Asheville Brewing Company are all good for a beer if you're passing by, but nothing super remarkable.
(Note: The Asheville Brewing Company location on Merrimon Avenue is super-kid friendly, with a game room, a movie theatre showing 3 different films during the day one of which is always kid-oriented and relatively cheap, tasty food: pizza, burgers, sandwiches etc).
River Arts District/West Asheville:
The Wedge-Nice ambiance, mostly outdoor seating, very good beers across the board
Noble Cider and Urban Orchard are two good cideries if thats something you enjoy
New Belgium- Tour is interesting, beer is what you expect with occasional one-offs
In Boone, AMB (Appalachian Mountain Brewing) is the place to hit, good beer and cider.