Yeah, you can toggle the shake in the system menu.
And for the original Game Boy Tetris song, I'm not sure, don't wanna get a C&D from The Tetris Company /j
Korobeiniki is a Russian folk song from the mid-1800s. Tetris holds the trademark for the specific arrangement of the song that they created for the 1989 Game Boy version of Tetris, but the melody itself is in the public domain. If you can find a freely-licensed performance of the song that isn't based on Tetris' arrangement, you can freely use it in your game.
Or you could try to find some other music that has a similar feel to the A-B-C songs on GB Tetris. Something like Understanding It by Cabel Sasser (I'm not sure Cabel would let you use that actual piece, because they use it as the ending music for the Panic Podcast). Something with that kind of upbeat style would be a perfect fit for Playtris. Whatever you use, you'll need to make it loop somehow.
--cmmartti (frosted-flakes on Discord)