I have completed all the puzzles and am just trying to get the collections, but have a question about the hospital. When I put a sick villager in the hospital it says recovering at the hospital. Do they eventually recover, or do you still need a healer to do the job?

You still need the healer,, thats just a place to congregate. so you can find them when you need to (if the docs dont go right away) that way you can bring the doc to them, they dont get better and if your doc doesnt get to them, they will die. this is a known _bug_ though, the docs not doing their job and im under the impression its going to be part of the upcomming patch.