When Leonid Gayday presented the film to the censors, he added a nuclear explosion footage into the epilogue, and refused to remove it, and only agreed after an extensive argument. This allowed him to sneak in some controversial (by the USSR's measures) material such as striptease, drunken debauchery and references to prostitution.
The handgun, loaded with blanks and given to Gorbunkov is a Beretta Model 1934, most likely a war capture. Soviet movie studios at the time were in possession of a lot of weapons captured from Germans, and were used in the war-themed movies.
The well-known quote of the film, "I won't be surprised if we are to find out your husband is secretly seeing a mistress," was originally, "I won't be surprised if we are to find out your husband is secretly going to synagogue." However, the Communist Party Committee of Ideology banned the phrase, because they felt it was "too anti-Semitic for the USSR". The phrase was re-dubbed, however, the original meaning can be easily spotted by looking at actors lips.
The quote "The fur coat can wait", said by Gorbunkov's wife, played by Leonid Gayday's wife Nina Grebeshkova, is based on a real event. Mosfilm gave Gayday two cars for filming and one of them he paid with the money he was originally saving to buy his wife a fur coat.