The songs are actually being sung by the characters in-universe. So how does Kathy know the words to join in on "You Are My Lucky Star"? Probably it's a song from The Dancing Cavalier, and she and Don have sung it before in character. (This also explains how Cosmo is able to jump in and conduct the orchestra: he would have conducted it during recording, and maybe even composed it himself.)
How come Don is able to casually climb a bus and jump straight into the passenger seat of Kathy's car? He was a stuntman. He's probably done many similar and more dangerous things on set.
