I'm not sure about else where but over here in Oz, it is the winners of the classes go in for Champion, which is awarded, then the person who came second to the champion competes against the others for Reserve Champion. Supreme Champion is out of all champions (pony, galloway & hack).
Yea, smaller shows generally only have two or three divisions in which they add up the ribbons from all the classes. The rider with the most points (1st places count for 6 points, 6th places count for 1 point) gets champion and second most points is reserve champion. It's obviously best to enter as many classes as possible in the division so you can have more points.
Thats how they do it around here but it's different all over.
Depends on where you are I think. At our schooling and A shows each division awards champion/reserve. We have 3 jumping classes and a flat class based on your horse (and 2 eq classes as well but they don't count for champion). Depending on your ribbon you get a certain number of points. Whoever has the highest number of points in that division gets champion, reserve is next. If there's a tie jumping classes count for more then the flat. But you can have a tie. I'm assuming those are the USEF rules.