I really like the idea of her having been around for a little while, working under the past HoH and then stepping up once she left!
As for dogs, there are over a thousand people living in the LBJ so resources are an obvious concern. I don't think anyone would tell her she COULDN'T have 3 dogs, but making sure that the people are fed is the first priority. It would definitely be at her discretion to make sure that three large dogs have enough to eat and that she takes them outside for exercise/to do their business within the time limitations that curfew presents. Another thing to think about is as HoH (or just a medical professional in general) is that she's in a position of power and would be advised to pay attention to her personal safety, since her skills are pretty important in these troubled times~ That being said, Savannah totally has a dog and has quite a soft spot for animals so Finn would be happy to have a companion (or two or three).