Regarding the song "Timothy", Joel was either wrong or just trying to console the Bots. "Timothy" was indeed about cannibalism that occurred between three trapped miners. Record studio executives had to sell the pitch to radio stations who banned the song that "Timothy" was an animal in order to get it played. The animal the execs used in their pitch was a mule and not a duck as Joel suggests.
In Segment 4, Joel, Tom Servo, and Crow refer to the Rupert Holmes song "Escape (The Pina Colada)" as simply "The Pina Colada Song" which is not the song's actual title.