Logging off a world while playing a music disc in the jukebox and then logging back in makes the music stops but the song notes continue until you either pop out the disk or exchange it with another one.
You can log off and in several times and the notes still keep going.
False, it still continues as long as the song would've been playing.
The sound is determined by the client, while behavior (redstone power, as indicated by the notes) by the server. If the sound stops (for any reason (logout, stopsound, reloading assets)) the server simply keept its timer going. If it didn' t, timing would be inconsistent based on actions unrelated to the block.
Not a bug.