Is it possible for diocesan priests to have a pet, like a dog or cat? If so, is it common?
Yes, of course, a priest may own a pet. In fact, Pope Pius XII had a pet bird. And it’s not just diocesan priests who can have pets. If the superior allows it, religious communities can have pets. A famous story in the life of St. Therese of Lisieux is that her mother superior was overly fond of the cat owned by the Carmel at Lisieux.
Just how common it is for priests and other religious to actually own pets, I have no idea.