A nice psychology article (media article) that provides answers to your question can be found at here. I will try to summarize briefly based on this article.
How close to uniformly randomness can we expect each person to act?
The short answer is no. As explained in the article, on the first round of Rock-Paper-Scissors it is less likely that a randomly selected person will choose paper over scissors or rock. Moreover, as two people play many rounds of Rock-Paper-Scissors together, they tend to not play randomly but adjust their selections based on whether they won or lost.
Will this be a stationary distribution or it will change over the time if there will be a lot of trials?
There are many definitions of stationary, but a paper mentioned in the above article found cyclical behavior which would violate some of the definitions of stationary. However, this might still fit into some definitions of stationarity.
While I'm unaware of a particular article which addresses the question of wide-sense stationary in this game, I would be surprised if this were found in Rock-Paper-Scissors played between humans. There is an interplay of affective (emotional) and rational decision making that tends to be influenced by the actions of human opponents.
I would be less surprised if this were found when a human plays a computer, especially if they knew the computer was playing completely randomly. People play different strategies in games played Human vs. Human as opposed to Human vs. Computer (e.g. the Ultimatum Game)