If you hit a dog and you don't know who the owner is then you are going to have to claim off your own insurance for any repairs (to the car, not the dog). This in the UK at least would be classed as your fault (because the end result was that all the cost of the claim was borne by your insurer).

I believe that if you know who the dog belonged to then in theory you (or our insurance company) could go after the owner to recover the costs. Dogs (in UK law) are classed as under control of their owners, meaning that the owners can be held liable for the dog's actions. Cats on the other hand are classed as uncontrollable and you aren't liable for their actions.

I now feel compelled to say, IANAL...