My DP is a chicken freak -- he calls it Vitamin Ch and can't go more than a day or two without any.
The words "leftover chicken" are a bit of an oxymoron around here! But I make a lot of:
Chicken and biscuits
chicken and dumplings
chicken pot pie
Chicken salad or plain chicken sandwiches
Chicken tacos, enchiladas, or fajitas
Amish-style chicken and noodles (I use the skinny kluski noodles or "pot pie squares" and just cook them in chicken broth with the leftover meat, some carrots, celery, onion, and peas)
chicken soup or tortilla soup or white chicken chili
I think that about covers it for my already-cooked-chicken repertoire.
As to your second question, no, not usually, 'cause I'm lazy. I stick the whole thing in the fridge, then the next day (or whenever) chunk off the big bits of chicken and then simmer the carcass in water for broth. When I take it out again I pick off the rest of the meat and discard the bones.