In the movie, Christopher Walken looks at his old master, James Hong, and says that seeing him was a "blast from the past", referring to his previous movie Blast from the Past (1999).
As mentioned in the movie, it is forbidden for Wong to teach ping pong to "gwai lo". This is most likely a reference to Dragon: The Bruce Lee Story (1993) starring Jason Scott Lee. Bruce was forbidden by the elders to teach martial arts to anyone other than Asians. Jason Scott Lee appears in "Balls of Fury" as Siu-Foo.
In individual Olympic sports, competitors are identified by their names, not by their nations (as many individual sports would have multiple competitors from the same nation). Randy Daytona's and Karl Wolfschtagg's names would have appeared on the scoreboard, not "USA" and "GDR," respectively.
In the scene where Randy and Ernie first meet Maggie, Randy rushes in with a hat rack to assist in combat but ultimately is disarmed by Maggie. When she backs him into the corner, and he's shrieking like a scared girl scout, you can see his left hand knocks a picture off its hanger, though not onto the floor. The camera then cuts away from him for a time, but when it returns, you can see the picture is again hung up perfectly.