During the scene in which Dan Aykroyd, Chevy Chase, Brian, and Stewie are brought before Russian Prime Minister Vladimir Putin, Putin takes out a hanger for his jacket amongst other things. This is a reference to a scene in Raiders of the Lost Ark where Toht takes out his jacket hanger.
The tracking device used to find Adam West is similar looking to the PKE Meter from Ghostbusters.
The plot of Brian, Stewie, et al searching for long forgotten Russian sleeper cells that can be 'activated' via a secret message is similar to Charles Bronson's 1977 film Telefon, which had the same plot device.
Vladimir Putin's description of mercenaries invading the Czech Republic is of the film Stripes starring Bill Murray and Harold Ramis.
The set of Stewie dolls popping one out of another is a reference to the Russian Matryoshka Dolls , also known as Russian nesting dolls or babushka dolls
When Peter was doing his John Wayne bit he mentions John Wayne Gacy. Gacy was a real life serial killer whom raped and killed an estimated 32 teenage boys. He was also known to dress as a clown for parties.
Quagmire's first suggestion for an improv troupe name "Improvda: The truth is ad-libbed" is a play on the Russian word for "truth," "Pravda." It also references a communist newspaper of the same name and so the troupe name comments on communist governments' propensity to twist or subvert the truth via the media.
The exercise in which Quagmire tells Peter to close his eyes and describe what he sees alludes to Dead Poets Society where Robin Williams' character does the same exercise with Ethan Hawke.