When Yuri Lushkov is taking a bow after his scene a short cut to the audience is visible. However Lushkov can be seen there as well, even if he is actually on the stage.
During the show Vladimir Zhirinovskiy, Grigoriy Yavlinskiy and Yuri Lushkov are sitting at their table in that order. Howver in the last song Zhirinovskiy and Yavlinskiy switched places.
The song "Tchip-Tchip", also known as the Duck dance, is credited as a french folk song. However while it became a classic, it is originally from Switzerland and it was written by Werner Thomas in 1957, so it is not really a traditional folk song.
At the end Putin is doing his New Year Speech from the studio of "Vremya", the News program of the 1st channel. However the set is from the end of the 90's and wasn't in use in 2002 anymore. However it is a running gag from another Pilot-TV show to use this set in any time period. As so it can't be considered as a continuity flaw.