The name of this episode is a reference to the Children's book The Cat in the Hat by Dr. Seuss.
Homer travels into the South Park universe in the couch gag. He also travels through (and boos) the universe of The California Raisins and their song, before encountering the Nerd from Robot Chicken and stealing back the painting.
The song that plays when Lisa is with the hat is Wouldn't It Be Nice by The Beach Boys.
Bart says that he is bad to the bone, a reference to the George Thorogood song "Bad to the Bone".
Donald Trump appears in Lisa's fantasy and he appears to be tweeting on Twitter.
Rod and Todd's nosebleed after levitating the compacted car through faith is a reference to Stranger Things. The character of Eleven in the series has a psychokinetic power which leads to a nosebleed after each use.
Mozart's Requiem in D Minor makes several appeareances while Homer reminisces about his past on chess.