The room with the North Koreans is room number 237, which is the haunted room in The Shining (1980).
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, and he is, indeed, making motions with his fingers akin to quickly but gently playing the piano.
The 'luxurious penthouse in New York's high rent district' where the action occurs is the 'Tuntmore Towers' - another part of the Tunt empire.
During a fight, Archer hits a North Korean agent and calls out, 'This is for the Pueblo!' This is a reference to the USS Pueblo, a US Navy research/spy ship which was captured by North Korean forces on January 23, 1968 and is still held by them.
When Archer and Lana are at the elevator, a North Korean agent comes off the elevator and kicks Lana in the chest. Archer quickly defeats the agent and checks on Lana. He comments, "Obviously it hurts getting 'Tom Dempsey'd' in the tits like that." He is referring to the former NFL kicker, Tom Dempsey, who held the record for the longest field goal (63 yards) for 40 years. Dempsey was born with no right hand and no toes on his right foot. He wore a special shoe that fit over his half-foot.
Five other kickers have 63-yard field goals to their credit, and one has kicked a 64-yard FG. (Two of those were kicked in the high altitude of Mile High Stadium in Denver, CO.) Dempsey's kick was a game winner as time expired in New Orleans on November 8, 1970. However, on September 26, 2021, the Baltimore Ravens' Justin Tucker hit a 66-yard field goal (at Ford Field in Detroit) as time expired to win the game 19-17 over the Lions and set a new NFL record.