Is there experimental evidence that people ever play mixed Nash equilibrium in real games? Have any studies been done that demonstrate people (not game theorists) actually using mixed Nash equilibrium as their strategy in a game? 
 A: According to this(article about mixed equilibrium strategies), I think penalty kicks between two soccer teams use mixed Nash equilibrium strategies.
A: There have been lots of studies on this sort of thing, with different results.  It depends a lot on cultural context.  You might look at "A Beautiful Math" by Tom Siegfried
A: 
This would be a fantastic question for a Game Theory StackExchange

My preliminary research backs up my intuition: it depends. Specifically, it depends on whether there is sufficient common knowledge to foster a gradient-like movement to mixed-equilibria.
Learning a Mixed Strategy Equilibrium in the Laboratory by Bloomfeld (1994)

Literature [on the evolution of behavior] raises doubts about the
  predictive power of mixed strategy equilibria, because very few
  mechanical processes converge to such outcomes. However, mixed
  strategy equilibria may have more predictive power when the feedback
  players receive is disclosed publicly. Such disclosure allows players
  to predict and exploit mechanical adjustments, and may therefore
  induce players to apply more sophistivated adjustment strategies that
  do converge to equilibrium.

An Experimental Study of Information and Mixed-Strategy Play in the Three-Person Matching-Pennies Game by McCabe, Mukherji and Runkle (2000)

Recent experiments on mixed-strategy play in experimental games reject
  the hypothesis that subjects play a mixed strategy even when that
  strategy is the unique Nash equilibrium prediction. However, in a
  three-person matchingpennies game played with perfect monitoring and
  complete payoff information, we cannot reject the hypothesis that
  subjects play the mixed-strategy Nash equilibrium.

How do People Play Against Nash Opponents? by Shachat, Swarthout, Wei  (2011)

We examine experimentally how humans behave when they play against a
  computer which implements its part of a mixed strategy Nash
  equilibrium ... A minority of subjects' play was consistent with their
  Nash equilibrium strategy.

