Twice in the episode, an announcer introduces "The Tonight Show" as originating "from Hollywood." While Johnny Carson (whom Oliver refers to as "the new guy") did take over as host in 1962, the program didn't relocate from New York City until 1972.
Response: Again, with Dancing Beene, just because we hear the song doesn't mean the characters do. It's just a song the writers thought would go with the scene.
This series is set in the sixties. So why is "I'm Too Sexy" by Right Said Fred--an early-nineties song--pumping on the soundtrack during the dressing-room scene?
The song was utilized in the context of an exaggerated flashback from a 2003 perspective. It wasn't portrayed as actually having been played in 1962.
The song "Dancing Queen" by ABBA wasn't released until 1976, although this series is based in the early '50s.