Yes dogs can be litter trained. They make specific boxes and litter just for dogs. I wouldn't allow a puppy to have access to a cats litter box....dogs happen to LOVE to eat cat poop. You'd just have some ill puppies on your hands if you did that. It does take a lot of work though, dogs are not as easy to train for a litter box as cats. My grandma tried it with her chihuahua but wouldn't put the effort into training him so she gave up. You would have to be home and put in the time to train them to use a litter box just like you would to train them to go outside.

I agree with others that a small puppy, especially more than one, would not be the best idea in this situation. Look into some rescues that use foster homes. If the dog has been fostered around cats and children and has been good with them that would be a fairer bet than a puppy IMO.