Bart says his birthday is too close to Christmas, but his birthday is revealed to be April 1 in The Bart Book. But in the season 26 episode Simpsorama he states his birthday as February 23rd.
The episode title is a reference to the Edward Scissorhands (1990) directed by Tim Burton and starring Johnny Depp in the title role. Homer also tells Marge to call him "Homer Fingerhands". The scene where Maggie twirls around under the falling hair being cut it a reference to a scene in the film where Winona Ryder's character does the same thing under the falling snow.
In the DVD commentary of S4 E8, "New Kid on the Block," it was revealed that the original B-story for the episode was about Homer being a great hair stylist. This would become the main plot of Homer Scissorhands nearly 20 years later.
When Patty describes Homer's brain as being the size of a super ball, we then see the inside of his skull and his brain bounces around and subsequently escapes out his ear to end up bouncing down the street. The sound effects used for this were actually from The Absent-Minded Professor (1961) as the sound Flubber makes when it bounces.