Paul Kinsey refers to Star Trek (1966) as a hit television series. In reality, the show had low ratings and only received a third season, because of a letter writing campaign by fans. The show's actual popularity didn't start until reruns in the 1970s.
When Don and Joan are at the bar, Joan plays Peggy Lee's "Christmas Waltz" on the jukebox. However, this song was never released on a 7" record that would have been in a jukebox at the time.
While Don and Joan are having drinks at the bar, Joan takes a puff from a cigarette that is not lit, and then lets its tip rest in the ashtray as if there were ashes to drop. We can clearly see the cigarette is intact and no smoke is emerging whatsoever.