Voices + Instruments

After a quick search on Russian sites I was able to find that a different actor is singing for Mels, the actors name is Andrey Birin. The actress that plays Polly did her own singing, so did Mels' father. Fred did not sing himself. In general, Russian movies do not hide the fact that actors voices are dubbed when singing and the names of the real singers are listed on the soundtrack for this movie.
Also, for some insider information, the songs used in the movie are all old soviet songs generally from the late 80s and early 90s and were very popular. The lyrics and the arrangement of the music was changed to fit the movie storyline, however all the choruses were original. This made the movie that much better to those of us who grew up in the Soviet Union during those years.

Looking at the way actor Anton Shagin puts his mouth on the saxophone, I suspect he doesn't know how to play the instrument at any level. His mouth is too far forward on the mouthpiece, and his lower lip is pushed forward instead of being properly tucked above the lower teeth. With his mouth in that position, I doubt he'd be able to produce more than a few feeble squeaks and/or honks on the instrument.

Zach Sullivan, who plays the spirit of Charlie Parker, commits the same faults. It's silly that they didn't get a real saxophonist to play that small part. A real saxophonist would also have been able to give Anton Shagin some much-needed on-set coaching.