There are a number of references in the episode to "Judge Crater." Joseph Crater was a New York Judge who vanished amid scandal in 1930; his disappearance would become one of the most mysterious missing persons cases of the 20th century. The case was never solved.
Carl Reiner had been a weekly viewer of "Leave It To Beaver." Three of the actors from "Very Old Shoes, Very Old Rice" played recurring characters on that series: Richard Deacon (Fred Rutherford), Madge Blake (Mrs. Mondelo), and Burt Mustin (Gus the Fireman).
Rob says, "(Laura) thinks I'm going to live forever." Indeed, Dick Van Dyke has outlived every adult cast member.
The music "Dodo" puts on the record player for the wedding ceremony is "I Wonder What's Become of Sally," a 1924 song with lyrics by Jack Yellen and music by Milton Ager. It's a song lamenting the loss of a woman, hence Rob and Laura's puzzled expressions after the music starts.
Buddy suggests a comedy bit in which Alan Brady would play a Cockney chimney sweep who is so fat he can't get down a chimney anymore. Dick van Dyke portrayed a Cockney chimney sweep in Disney's Mary Poppins, released in 1964.