can you litter box train a dog?

I suppose you could but would you really want to? It would REEK.

No. Dogs need to go out for exercise, potty, and mental stimulation.
It may be possible, but would not be good for the dog.

Mine don't go out when it rains or when it is too hot. One of them is a Peke and he could get a heat stroke easily. They both know where to relieve themselves inside but they prefer to do it outside of course.

They always do everything they can to encourage me to take them out. They also love to ride in the car so we sometimes go to parks miles away. I do make up for them if they can't go out for a couple of days.

Yes you can. However it is not recommended. As stated by another, your dog needs exercise. Dogs are evolved from wolves, who are nomadic and travel good distances. It's still in their genes (even if their current physical forms are a bit more slothy), and that's what I recommend. Good old fashioned walking around with a baggie in your pocket.

If you want your dog to learn its ok to go in the house. Otherwise train them to go outside to potty.