When Ivan sees Katya in the crowd trying to see Stalin's corpse, he crawls over the throng to get to her (which is so thick people were literally crushed to death), but then is able to jump down, pick her up, and run with her to safety through that same throng.
When Hulce's character is shown Stalins projection room he announces "You have Mitchells!" referring to the make of projector. No such manufacturer exists. Mitchell was (and is) an American manufacturer of cameras, not projectors. There were Soviet makers of theater projectors, but no Mitchell.
The movie covers the years 1939 to Stalin's death in 1953.
However, it constantly refers to the KGB, an organization that did not come into existence until 1954. Until then the USSR security service was known by a variety of names, most notably the NKVD (Narodny Komisariat Vnutrennykh Del or People's Commissariat of Internal Affairs) between 1934 and 1943. It is this organization that Ivan would have joined in 1939. However, the sign on the NKVD headquarters in the film does read "NKVD" not "KGB". This refers only to foreign versions of movie, because in russian version names are correct. "NKVD" is used in early scenes and "MGB" in '50s.
The moment than Ivan Sunchin has a meeting with his NKVD (KGB) superiors in Moscow's "Revolutionary Square" Metro Station. Just before coming in to his superior bunker office, he is strolling along the station. The plan of Arbatsko - Pokrovskaya line is visible on a wall of the station . The plan consist of five or six transfers to other lines of the Moscow metro, which was current for Moscow metro circa early 1990s, not 1940s than movie plot set. In early 1940s Arbatsko - Pokrovskaya line had only two transfers to other lines, and metro plan looked different.
Ivan refers to German troops being so many miles from Moscow, but both Germany and Russia measure distances in kilometres.
When Anastasia asks directress, if the Pokrov town located to the south of Moscow, she (maybe intentionally) answers 'Yes'. In reality, Pokrov is located to the east of Moscow.