Old blankets are your friend :) My car's backseat is covered in blankets…if they were to make a mess I can toss those in the washer.
Oohh.. Pretty similar story with my cairn terrier. He is cute and adorable but once we start to go ride on our car, he acts weird. We don't bring crate. Just like this one, we bought a pee mat(2 pcs), one at the flooring and one at the next to my seat since he used to be with me always. We do that since it was untimely when he pee, maybe because of nervousness.