Well, it is kinda tough to ask for money for your car when you killed their dog with it.
Doesn't matter. Leash laws would mean the dog should have been restrained and they're responsible for the damage to the car.
I wouldn't be able to do that, I'm too inclined to look at things from the other guy's point of view. Someone just killed my dog and now wants me to pay them for doing so? Nuts to that!
I'd probably just tell the insurance company I thought it was a deer. That's normally covered as a comprehensive event, which means no rate increase regardless of fault. It would be more complicated with a police report involved, though.