Sharon,
We just stayed at Frenchville House B&B in Galway. https://www.frenchvillehouse.com/
It was one of the most incredible stays I've ever had. Pam and Tony are consummate hosts. The breakfast (included in the room price) is fantastic.
It's not exactly a family room like you're looking for, but I plan to stay there again when I travel back to Galway with my kids (ages 16, 14, 6) and husband). It's a home. We stayed in a room at the end of the hallway, and there was another room directly across. Would provide the same feel as a family room.
Not right in the city center, but very accessible. We took a 10E taxi, Pam scolded me for not asking for pickup from the train station. It might be walkable, but it was rainy and my I'm traveling with my mom who is spry but not into huge walks. There is a huge park and walking path right across the street.
For your AC question, we were also in the UK on this trip, stayed in a cottage in Mildenhall village with no AC. It was quite hot during the day (for there), about 25-30C (75-90F). If we would've been in the room during the day I would've wanted AC. However, it cooled right down at night. Galway was much cooler, even than Dublin. I think you'll be fine without AC unless you plan on being in the room during the afternoon (nap time?).
Cheers!