Camped with dogs for decades, several generations of dogs; Barking has never been an issue (they were pretty well trained and socialized).
I have noticed that some dogs seem to pickup on owner fears, perhaps contributing to some barking problems.....
Our dogs would regularly sleep under the vehicle in hot weather (only shady spot in the desert); yes they do occasionally get greasy; spot clean with dish soap and water (then fix the leak/drip when you get home).
Mostly they liked to sleep, at night, in the ground tent with the rest of us; don't zip the door all the way closed so they can let themselves in/out. Bring their bed/rug etc from home and give them a spot in the tent, on the floor.
Hunting breeds sometimes take off after deer/elk but so far they have come back after 3 or 4 hours.
Note; we do not camp in "civilized" areas that require reservation or have; rules (apparently only for people to ignore), numbered sites, showers, toilets or other "amenities".
Enjoy!