Answer:
Rabbit - Butch; Turtle - Pete; Dog - Star; Cat - Mike
Step-by-step explanation:
1. The rabbit and the cat gave birth, so they are females.
2. Mike is a mother, and the rabbit is not Mike. Mike is the cat.
3. Pete is older than the rabbit and the dog. Pete is the turtle.
4. Star likes to have his back rubbed. That's not an important piece of information, but the use of the word "his" is. The rabbit is female, so the Star would be the dog.
5. Thus leaving Butch as the nickname of the rabbit.