I'm in Houston, and the Texas standard seems to be lots of mustard. And often jalapenos.
Myself, I'm terribly open about burgers. I generally prefer American cheese (but it depends on the burger), and I usually add lettuce, onions, tomatoes, mayonnaise, mustard, and ketchup (although I can live without one or the other when it comes to tomatoes and ketchup, which makes sense). I don't bother with pickles, but I like them on burgers, too (if that makes sense, and it probably doesn't). I also like bacon sometimes (it depends on my mood or hunger level).
As for all the other varieties, that's another story, and it often depends on the restaurant, etc. I'm learning (too slowly) that I'm really not into barbecue sauce and onion rings on burgers (especially from BK--yech).
Vegetables must go on the top. Cheese can go on the meat or bun. Ideally, the bun is toasted or at least warmed.
When my mother makes burgers, by the way, she puts the cheese on the bottom bun and the meat on the top bun to serve it. It drives me nuts. I always have to pick the meat up and put it on top of the cheese.