Your dog probably needs to be monitored more closely for now. It's possible that he was confused by the cat box - I once had a dog who taught himself to use the litter by watching my cats - and I absolutely agree that a dog will sometimes use infrequently visited rooms within a house as a toilet. It is as you described, they just don't recognize these areas as part of their "den." Also, it's not unusual for a new rescue to be on "best behavior" for the first few days or weeks. Issues sometimes don't reveal themselves until the new pet feels truly at home.
You have gotten a dog, an adult cat, and a kitten in what...less than two months? That is a lot of change for all of them. They not only have to get used to new people, new surroundings, but also to each other. Obviously, you will have the least trouble with the kitten, but with the adult animals, you are probably just going to have to be really patient and, above all, consistent.
Don't you have another thread about your older cat being a little head-shy about petting? Same answer: patience. The cat is doing well to not be completely freaked out at this point!
I'm glad you are not resorting to physical punishment. The only thing that will accomplish is to destroy your pet's trust in you. Actually, the best reaction to your dog's mistakes is none at all. No yelling, no fussing, just clean it up as wordlessly and silently as possible. The way you have described his reaction to the dog door makes me wonder...is it possible that he is fearful of being alone outside? I would keep an eye on him at all times in the house, crate him when you must leave for a bit, go outside with him, and praise him to the skies when he eliminates outdoors.