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 was just inaccurate.