Batman. Why? Batman always wins.

Sheldon Cooper VS Shawn Spencer.

Batman because he once took on the entire justice league when averaged out by a Thanagarian virus and his training and discipline still let him win.

Stupidity vs Intelligence

Stupidity. It's infinite, unlinke intelligence.

Again: Sheldon Cooper VS Shawn Spencer (you skipped over it)

Shawn Spencer. Because at the end of the day, he's happier, has actual friends, and romantic relationships that aren't nearly as cringe-worthy as Sheldon's. (And if we're talking a traditional fight, Sheldon has all the muscle tone of a wire hanger, and no coordination whatsoever. Shawn's at least normal-to-upper level slacker fitness and fighting skills, I think.)

Richard Nixon or Chairman Mao. Assume both to be in their "prime".

Chairman Mao
Nixon still has to deal with Watergate, which is his downfall, while Communist China is still going strong.

Deadpool vs a Gaming Group's Toon campaign

Mao. More cold blooded. He was waaaaay more willing to send people to their death. Nixon was a creep, but I don't think he had mass murder in him.

film version Wolverine vs. film version Captain America

Film Version Wolverine. He's nigh on immortal

Deadpool vs a Gaming Group's Toon campaign again, since it was skipped

I'm going with the toon game. Better writing. Better acting. More original. And a far superior plot.

Speaking of.

Darkwing Duck vs The Tick.

Darkwing duck, as I don't know the tick.

Greek pantheon versus D&D pantheon.

D&D pantheon, but only because the Greek Pantheon was also in a DnD book, thus making them a part of the DnD pantheon (you can't fight against yourself). Though even without that, there are a lot more DnD gods and their power is usually displayed as being stronger than the depictions of the powers of the Greek Gods.

Xykon vs. Voldemort