Adam's recruitment poster for the Mel Brooks fan club shows Dark Helmet from Spaceballs requesting members on order of their "Stupid President", which gets mistaken for an anti-Republican political cartoon. If that's the case, then why didn't anyone notice that the depiction of the president looked a lot like Darth Vader? They did, but they thought the depiction was a reference to the Strategic Defense Initiative, often derisively referred to as "Star Wars" (which came up later in the episode).
Both Pops and Beverly have a habit of injecting Yiddish into their dialogue. Since Beverly is Pops' daughter, this is probably a thing from their side of the family.
After Adam uses all of Murray's stamps to inflate his chances of winning the Toys 'R' Us contest (and loses), Murray tries to punish him by breaking his toys. Specifically, he tries to rip apart Stretch Armstrong, knocks the head off of a Rock 'Em Sock 'Em Robot, and throws some Wacky Wallwalkers against the wall. Adam points out that that's what you're supposed to do with those toys, so he's actually just playing with them. Basically, he's doing with Adam's toys what Adam did with his stamps: using them for their intended purpose.
Coach Mellor repeatedly calls Adam "Goldfarb", most likely because he can't picture geeky Adam being related to jock Barry, whose name he does get right. But why "Goldfarb"? Well, remember when Mellor had to teach math and he showed the class Cannonball Run II because it had a number in the title? If he's a fan of the Cannonball Run films, he probably got it from Roger Moore's character in the first film.
Despite constantly portraying Barry as a moron, the show likes to remind the viewers that Barry later became a doctor. Since Barry's brother Adam is the show's producer, the reminders that Barry became a doctor can be seen as Adam being proud of him.
Barry's friend group calling themselves "JTP" for "Jenkintown Posse" makes more sense once you learn that the school William Penn is based on is several miles south of Jenkintown's city limits, so it's possible they're the only students in their class from Jenkintown.
In "DKNY", Adam thinks Dave Kim's British friend Tamsin is Australian at first because he had ""Crocodile" Dundee on the brain". It later turns out that Tamsin is actually from Tucson and is faking being British. So maybe she didn't get the accent quite right earlier and that helped steer Adam towards thinking she was Australian.
