I agree with your other picks, but let me just state my opinion on KOTH's best episode.
King Of The Hill - A Firefighting We Will Go
This, in my opinion, is the defining episode of the entire show. In this episode we get a Rashoman style viewing of the events that transpired over the previous few days from the viewpoints of all four of the main characters. This gives us a real in depth look at who they really are and what they think of themselves and other characters. The narration from each character merely discusses the events, but if you pay attention you can see little details like the color of the environment, or subtle appearance of character that differs from reality. It's very fascinating to watch.
But, on top of that, it's really damn funny. You see exactly what happens when the guys are given free reign over something, and not only are they incompetent without Hank, even Hank himself is shown as powerless without his friends.
It's the funniest episode of KOTH, easily. But it's also got a lot of great character moments. Definitely the best episode.
As for characters, the best episode for each character is:
Hank: Nine Pretty Darn Angry Men
Dale: Soldier Of Misfortune
Bill: Pretty, Pretty Dresses
Boomhauer: Patch Boomhauer
Peggy: Lupe's Revenge
Bobby: Bobby Goes Nuts
Luanne: Buckley's Angel
Another great, though pure comedy, episode is Three Coaches & A Bobby. I really wish AS would show the 3rd season.
Agreed. I think it's a tie between Super Fun Time and Tsst for my favorite SP episode.
In Super Fun Time, every character shines, it's got that wonderfully over the top-ness South Park is known for, and the fact that there's no real B-story. The B-story is simply another group dealing with another aspect of the A-plot, which I really like.
Plus, the Die Hard-esque villains robbing a Burger King is such a great reveal.
As for Simpsons... That's a hard call. For humor, I would pick Homie The Clown. It's a simple 'character swap' episode, but to the extreme. Plus, my all time favorite scene- "STOP! STOOOOP! He's already dead!"
But in general, probably Bart Sells His Soul. The main plot is pretty emotional, and the B-story is hilarious, so you get a nice balance of both. I would definitely show this episode to get someone into the series.
Futurama is easily Roswell That Ends Well, as it has everything Futurama is known for. Great characters, unpredictable situations, and manages to be really funny at the same time. For emotion, I would pick Jurassic Bark, as it manages to have a real punch most shows don't have.