In "A Song For Margo" Jay reviews an ET The Extra Terrestrial parody called "D.T. The Drunken Terrestrial." D.T. is also an abbreviation for withdrawals alcoholics go through after they stop drinking.
Double Fridge Brilliance: In "All The Duke's Men" Duke runs for president and is campaigning to a group of zombies, where he see Doris who's not dead, but looking for a husband. She has an unrequited crush on Duke, so she was following him and trying to get his attention. The zombies was another way of saying that American voters are zombies and vote without doing research on the candidates.
In the second season Jay Took a Level in Kindness. The theme song reflects on this. In the first season it was slower paced, and Jay was crankier and more quick to anger. In the second season the theme song is more upbeat.
In the pilot episode, Jay meets an actress who pretends to be in love with him so he'll give her a good review. Her acting in the movie is atrocious, but she manages to convince him. She really is a good actress, she was just given a crappy script and the film more than likely had a bad director. This also happens again in "From Chunk to Hunk" with another actor who Jay bashes, then they convince Jay they're gonna kill him, which turns out to be acting and Jay changes his mind.
The XYZ Men parody in the web series is a continuation of the XYZ gag in the early episodes of the original series.
Why would someone like Jay become a critic? He's obviously a nerd and cynical, he's made a career criticizing stuff, but since he was unpopular growing up and was adopted, films are a way for him to escape reality for a while.
Jay's obesity is more of an Informed Flaw and he appears to be the size of and looks like an average man. Keep in mind he's in show business that has high standards when it comes to appearances.