No offense intended to the OP, but these requests are asking for a gun to do a lot of conflicting things. It would be like asking for a rifle to shoot F-Class, 3 gun and wanting to take it out to hunt moose and gophers a couple of times a year. The surprising thing is that you can come close.
If your budget is $800 max forget the O/U. You may get a gun that is OK, but you may not. Too much of a crap shoot. You need to stick to a semi-auto, and in fact, may need to go used there to get a quality one. Look for a 28" barrel with screw in chokes. It will get the job done on all you mentioned without major limitations, just won`t have the specialized features that separate a trap gun from a skeet gun from a waterfowl gun from an upland bird gun.
Good luck!