The episode title, "White Nights," is a reference to the 1985 motion picture "White Nights", starring Gregory Hines and Mikhail Baryshnikov. This can also be a reference to White Nights - phenomenon when during northern summer evening twilight is followed by morning twilight and full darkness never happens. Although such thing does not happen in Moscow, but widely associated with St. Petersburg. It could also be a reference to a short love story by Fyodor Dostoevsky. White Nights is also the name of Menachem Begin's memoir of his year in a Soviet prison camp after being arrested by the NKVD in Lithunia while fleeing the Nazi invasion of Poland. Given Adam Reed's frequent literature references, Dostoevsky's short story seems right, but Archer's brief stint/torture in KGB custody links to Begin's memoir. Given how tightly Archer is written and produced, it wouldn't be surprising if the title references both.
The submachinegun Katya Kazanova uses is a PPSh-41, a World War II-era Soviet weapon, almost universally considered one of the best submachineguns designs of the war period and widely distributed to Soviet-aided insurgents and revolutionaries during the Cold War. The sound effect used, however, is inaccurate, as one of the most recognizable traits of the '41 is its fairly high rate of fire, about 900 rounds per minute.
Moscow skyline is pictured incorrectly, showing St Basil church from a place where it can't be seen and overcrowding skyline with churches and spires, most likely as a parody of practice of using well-known landmarks to establish locations.
Word "Директор" on Jakov's door means "Director", but KGB chiefs are actually called "Chairmen"