I live in NYC and all the dogs I know travel in elevators without relieving themselves. Most (adult) dogs in my building go out three times a day and do just fine. "Sit" in the elevator works great until you have the elevator rules established (and it's good to fall back on when you have an ancient dog on Lasix).

Ten hours alone is too much for many dogs, though day care or a dog walker could help. The cat issue sounds harder unless you find a huge apartment.

I think returning the dog to the rescue is acceptable, responsible and probably best for everyone. However, you definitely should tell the rescue the dog demands to go out every 20 minutes and in the middle of the night. Perhaps now that his health issues are resolved he could get some additional training.