Apparently Cheryl has developed a taste for animal tranquilizers, which she steals from the veterinarian's office at the zoo while he is at lunch. While chastising Pam for voyeuristically watching he and Lana have sex in his office, Cyril specifically states that Pam needs large pills, at which point Cheryl reveals that she is interested in some too, mistakenly believing that he means the tiger tranquilizers she enjoys.
Cheryl also says the ultimate aphrodisiac for her is making love on top of a tranquilized tiger.
While receiving a massage, Pam complains to her masseuse for not being firm enough, saying "Who are you? Van Cliburn? Knock off the ticklin' and work that shit!" Van Cliburn was a famous piano player. "Tickling the ivories" is an idiom referring to the act of delicately playing the piano. In fact, Pam's masseuse is positioned such that Pam's back is like a musical keyboard or piano, and he is indeed making motions with his fingers much akin to quickly but gently playing the piano.
Archer refers to the superstition that cameras steal souls, ascribing it to Koreans. A North Korean agent rebukes him, saying this is a belief of Native Americans, which Archer hypocritically calls a racist remark. In reality, this is a superstition held by people in many parts of the world including some Europeans, although there are similar superstitions which are incorrectly translated.
Once again, Malory shows one of her grudges from World War II. In this case, suggesting to a French waiter he should apologize for Dunkirk. She probably does not believe the French fought hard enough or did enough to relieve the forces which were trapped there. She has previously alluded to the collaborationist elements in France  and the neutrality of countries such as Ireland during the war.
This is the first episode in which Malory's rival Trudy Beekman appears, after being mentioned numerous times. At lunch, Malory is mortified when she reaches for her ISIS Black Titanium Credit Card, only to find it is not there because Archer "borrowed" it for the mission. Trudy Beekman is delighted to show her up by paying with her Platinum card. We do not see her face because her back is to the audience, but we do hear her voice.
Apparently Pam likes gyros, as she is seen enjoying one in this episode. She had previously referred to "Arab hoagies" - this may have been what she meant. Another word for "hoagies" is a "hero sandwich," and the gyro is often pronounced "hero." Although they are often served in Greek restaurants, they or similar wraps are also popular in Middle Eastern and Turkish cuisine.
Lana is once again screaming lines from Full Metal Jacket as she mows people down with machine gun fire,
Lana finds herself once again re-examining her life and her relationship with Archer, ISIS, etc. She starts thinking about how she could finally do something that gets to Archer and seems to consider going through with it, which she decides in favour of when Archer keeps escalating the situation. She may have been considering whether to marry Cyril. One point in favor of this decision, according to her, is she was not getting any younger. And it would definitely make Archer upset. Another clue that suggests marrying Cyril was the decision she was contemplating is her reaction when she finds out he was once again insecure about her being with Archer and spied on her, she tells him that she has now unmade said decision. It is also possible that she was considering sleeping with Archer, which would get Archer back "on the hook" for her and also get back at Cyril for his distrust of her. (You can read Lana's character page for an end-of-season spoiler. )
While fighting one of the North Korean agents, Archer mentions the USS Pueblo, a Navy Intelligence ship that was captured by North Korea in 1968 and currently serves as a museum in Pyongyang.
The room in which the fight with the North Koreans takes place is Room 237. Room 237 is a reference to the creepy room at the Overlook Hotel from Stanley Kubrick's film version of The Shining (in Stephen King's novel, it's Room 217).