I've had two cat vs. car experiences.
When we lived in Pensacola, on the Naval Air Station, TCS would pull his small truck onto the grass yard to wash it. When he was done, he'd drive it back to the parking area. That was standard in our housing neighborhood.
One time, when he was thru washing the truck, he started to drive it. It made a thump, so he stopped. Horrified, we saw our kitty run away with a bloody face. We eventually found him, hiding behind the air conditioner equipment. His jaw was bloody. We wrapped him in a towel and drove to the vet's office. We had called him and he said he'd meet us there (it was a Sunday afternoon). Our kitty had to have his jaw wired, and spend the night at the vet's office. He eventually healed OK.
We figured out that kitty had hidden in the wheel well of the truck. When TCS drove forward, kitty rolled forward and under the tire. Because the grass was soft under the tire he didn't get killed.
My second experience was driving our big old Oldsmobile for car pool. I drove our daughter to school, and picked up some other kids along the way. As I was slowing in their neighborhood to stop at their house, I saw a kitten at the side of the road. It looked like it was going to cross in front of me, so I stopped. And waited, and watched for it to show up on the other side. Nothing. So, I turned off the car, got out, and looked around. No cat. Where did it go? I popped my hood and looked around. I could hear meowing in the engine compartment but I couldn't see it. I knew that I couldn't start the car again but I couldn't find the kitten either. Eventually, a police car stopped by. He called Animal Control. They arrived, put on long, heavy duty gloves and started feeling around in the engine compartment. They finally found the kitten, and we were able to continue on to school. I had to go into the school office to explain why we were late.