We have a cat along with 2 dogs and they all love each other! We got our cat, Annie when she was a kitten, so she's grown up around dogs. She was a little weary of Bailey when we got him (she's almost 2 years old now) but that lasted for maybe 2-3 days.
About the litter box, that was the problem we had. When Dozer was a puppy he would get into her litter box & eat her poop
It was the most disgusting thing, especially when he'd try to come give you kisses right afterwards. We installed a cat door and she goes outside to go potty, we don't have a litter box anymore. As far as I know, most dogs do like kitty turds....