Research shows that wearing your mask, and opening the front passenger window, and the window behind the driver seat helps to reduce the spread of the novel coronavirus in your vehicle. See this NY Times article.
I have a service dog. A 62 pound black Labrador retriever. Very sweet. I have a seatbelt for him that attaches to any human seatbelt. I'm happy to bring a towel or something to put under him in case he sheds.