The song playing in the Button Club at about 35 minutes in is a Chinese language version of the Gershwin brothers' 1937 hit " They Can't Take That Away from Me".
The set for Sophie's game show is modeled to look like the second set from the 1961-1967 CBS version of Password (1961)(Password).
The term cut contest, more commonly known as a cutting contest, comes from the world of jazz. It refers to when two musicians, usually playing the same instrument, alternate choruses, each trying to outplay the other. Its use here refers to the similar manner in which Midge and Sophie swap insult jokes during the warm-up.
(At 29:30) Midge says to L. Roy Dunham "You are a woman" "And you have always been a woman" to which L. Ron Dunham responds "I have yes, been a better story if I hadn't!" and winks. It is a nod to the fact that Hari Nef who portrays L. Ron Dunham is transgender.