Regarding Jay's appearance on The Simpsons..Jay was a guest star on The Simpsons. Jay exists in The Simpsons universe, hence why no else one from The Critic appeared. Jay's not a Butt-Monkey in his appearance which was probably made after he met Alice or is a case of inverted Adam Westing.
Jay's boss Duke Phillips is a Depraved BisexualHe's the one who has a crush on Jay and makes jokes about Jay to cover up his insecurity. He's from the South where homosexuality isn't as tolerated.
Doris really is Jay's Mother.The DNA Test results were switched or of poor quality.
Every movie in Hollywood since 1994 has taken a page from the show.Many jokes on the show came true in real life. The show had a bigger cult following than we know and any bad movie that's come out or sequel they predicted was due to this show.
Duke knows Noah Webster is dead.A famous scene in this show involves Duke making the word "quyzbuk", and when Marty calls him out for it, phoning Noah Webster to bribe him to make it a word, supposedly ignoring Noah being dead for who-knows-long. However, it's entirely likely that Duke knows this, and was just playing so that way Marty (who's young and gullible in this scene) will shut up about it.