I had rentals for years as well as managed rentals for others. I am a dog and cat lover, always allowed dogs but found that there needed to be guidelines, not everyone is a responsible dog owner. Cat were another story, you can not get cat smell out of anything, not even hardwood floors, so we were far more selective with cats.
Dog we insisted that they had to have all there shots, must be fixed unless specifically a breeding dog, did not allow dogs to be tied, must provide shelter for any dog allowed outside. We lived in Northern New York (Lake Placid) and the number of people who would put a dog out in the cold was unreal. This did not eliminate all problems but the people who meet these criteria were in general dog lovers who cared for their animals.