Have any studies been done that demonstrate people (not game theorists) actually using mixed Nash equilibrium as their strategy in a game?
According to this(article about mixed equilibrium strategies), I think penalty kicks between two soccer teams use mixed Nash equilibrium strategies.
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
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.