Thanks for the info! That's a very lovely song. I found a longer version of the video that has more of it at the beginning before they launch into the Rammstein cover. No wonder some of the audience looks so bewildered - the song did not turn out the way they expected!
Tchaikovsky does use snippets of traditional songs in his overtures sometimes. For example the 1812 Overture has at least two of them. Actually I think it might be three, and the French national anthem is in there too. One of the traditional songs sounds somewhat similar to the one we're talking about right now but isn't quite the same.
So the melody we're talking about might be Tchaikovsky AND a folk song. I think I've played it in orchestra as part of some huge symphonic piece, but I can't remember what it was.
The black squeeze-box instrument with buttons on it is an accordion. The round stringed instrument looks similar to a mandolin, but I'm not sure if that's really what we should call it, or if it's a traditional Russian instrument that doesn't have an English name. Are the triangular stringed instruments balalaikas?