I have a real problem with cats catching birds. I have cats that go outside into the woods and they hunt a lot. If I catch them with a bird I will take it away from them. Many times it is still able to fly away. They catch lots of rodents and I have no issue with that (as long as the body stays in the woods)
I always have to patrol my backyard to make sure there are not dead rodents or birds out there before I take Gracie out in the yard. If she sees a dead body on the ground she goes right for it and I don't want to have to grab it out of her mouth.
Your dog is naturally going to be interested in animal sounds. You just don't want the dog eating any of them!