Awesome, this all sounds good to me for both Liv and Pete!
I don't know if there is a limit on the number of pets someone could have at LBJ, but Savannah and Nick both have dogs and there used to be a character who served as a 'vet' for the LBJ and other shelters. The previous HoH had a dog as well. Not sure if 3 would be too many, but at least 1 is definitely possible based on previous established stuff!