Yes, the miniature schnauzer is in the AKC Terrier Group

No, there are 3 types of schnauzers. Miniature, Standard, and Giant. They all vary in size.

The Miniature Schnauzer is classified as a terrier while as the Standard Schnauzer and the Giant Schnauzer are classified as working dogs.

Well, first I have to tell you something. Miniature Schnauzers were meant for ratting, or to kill rats. Well obviously the rat will die but your Miniature Schnauzer will be fine.

--- As an addendum to the point above, your Miniature Schnauzer may not be fine - rats may be rife with bacteria and disease, which your Schnauzer may not be used to. If your Schnauzer has bit a rat, it may be an idea to discuss it with your vet, as the bacteria could cause your Schnauzer to get sick.

Yes it did