I'm sorry, that's horrible. Did anyone else see what happened and stop? If so, chat to them first and see if they'd be comfortable backing you up on what happened (dog ran out, no time to react). Hopefully that's not needed, but you never know.
I saw that exact thing happen when the car in front of me hit a dog. Poor thing was dying on the road and I heard someone down the street whistling for their dog to come inside. I went and told them what happened, and made sure to say that the driver is distraught and absolutely could not have done anything to avoid the dog. Fucking sucks for everyone involved
The answer to your last question should tell you why everything you wrote above it is fucking awful.
If you're driving safely and an unleashed dog sprints into the road and goes under your car from the side, what the hell do you think you would do to avoid it?
It's the owner's responsibility and fault, not OPs