After watching Bart's ad, Bart remarks that he "doesn't remember ever being in a commercial," then pulls out a Butterfinger bar and holds it to the audience for a moment before eating it. This is of course referencing the fact that Butterfinger dropped their contract with The Simpsons a few years prior, and Bart already seems to have forgotten working with them, while still obliviously plugging the product.
The doll used in the courtroom scene where Bart is asked to show the court on the doll where he was abused (punchline: the pockets) was a Bart Simpson doll. Generally, a generic, faceless doll is used in such situations because a) it neutralizes the emotional state for the defendant and b) because most children do not have personalized replica dolls of themselves, yet since The Simpsons manufactured Bart Simpson dolls the piece of merchandise subtly fit back into the show.
The scene in which Homer is sitting at his computer with a large glass of whiskey, watching Bart on the live feed from Tony Hawk's webcam, and says "Of all the sites, on all the web, I had to click onto his..." is a reference to a classic scene in the film Casablanca, in which Humphrey Bogart's character, Rick, sits at a table in his bar, drinking a large glass of whiskey, and says, "Of all the gin joints, in all the towns, in all the world, she walks into mine..." The music that starts playing at the end of Homer's line is "As Time Goes By", which is played several times in Casablanca, including during the aforementioned scene.
The title of this episode comes from the title of the 1979 Burt Reynolds movie, Starting Over, which was the first film James L. Brooks worked on.
Wanye's World is referenced when the Kennedy centers honor recipients say not worthy! to Lisa when she walks by.
When Homer shows Bart the pictures of him dangling Bart over the edge of an apartment, it's like the photo of Michael Jackson dangling his baby son over the edge of an apartment and dropping him. According to the writers, this had been added in last-minute, largely due to it occurring recently.
The midair skateboard battles at the episode's climax resemble the style of fights in Dragon Ball Z.
Blink-182 is playing their hit "All the Small Things". Their cameo in this episode is presumably due to the fact that their music was often used during the skateboarding fad of the 90's/00's era.
A piano cover of "As Time Goes By" is played when Homer is watching Bart on the webcam.
This episode was advertised to be the 300th episode of The Simpsons, but, in production code (i.e. broadcasting) order, it is the 302nd episode.
This is referenced in the episode itself. When Homer challenges Tony Hawk to the skate-off, Marge asks how many times Homer has done something crazy. Lisa then pulls out an actual counter that she clocks over to 300. Marge questions the number, saying she "could've sworn it was 302".
This is also referenced during the title screen in the opening sequence. Underneath the title are the words, "300th episode", even though The Strong Arms of the Ma was actually the 300th episode to air in broadcast order and Barting Over was the 302nd to air in broadcast order.
Originally, the reason Homer had squandered the proceeds of Bart's ad money had him buying a star that went supernova, and he wasn't able to get his money back. However, this was changed at the last minute to him being subject to blackmail due to the then-recent occurrence of Michael Jackson dropping his baby while presenting him. Something similar happened with South Park.
In real life, you have to be 14 to 17 to be legally emancipated from your family--although Judge Harm herself did acknowledge that while Bart may be too young to be emancipated from his family, she was willing to make an exception, since even she felt that Bart's probably better off living on his own than with someone like Homer.
Bart is clearly living on the top floor of the loft building (as indicated by the pan-in from the Absolut Krusty mural painted on the building's exterior), but when he panics and takes the elevator to escape the rat in his room, the elevator takes him up - numerous flights - and opens at Tony Hawk's pad.