Were films such as Dr Strangelove played in the USSR? And, as a related question, what was the opinion of the state (or the KGB) on such films?
Dr.Strangelove was banned and never screened officially in the USSR, most likely because premier Dimitri and ambassador de Sadesky were shown in a very negative light as compared to the POTUS.
source: list of all foreign films sceened in the USSR
However there was some domestic cinema and literature devoted to the cold war and the prospect of mutual destruction.
There will come soothing rain is a cartoon based on a short story by Ray Bradbury. Letters from a Dead Man is a post-apocalyptic science fiction film based on a novel by the Strugatsky Brothers. The Inhabited Island (translated, afaik, as "Prisoners of Power") is another novel of theirs which shows cold war backed with nuclear weapons on another planet.