The music played during the confrontation between Jesus and Santa is from Mortal Kombat 3. More specifically, the continue screen music (playing during the "verbal battle"), the Fatality music (playing when Jesus and Santa shout their initial "battle cries"), and finally the background music from the Subway level (during the actual fight).
The word "fuck" is used roughly 18 times within the 5 minute short. While 18 may not sound like a lot, this is roughly an average of saying this word once every 17 seconds - even more than Casino (1995)'s average of every 25 seconds and Scarface (1983)'s average of every 49 seconds.
This short was made for FOX executive Brian Graden to send out to his friends as a Christmas card. Because of some of the content he realized when it was completed that he really couldn't send it out to all his friends. The ones that did get it, passed it around and the resulting buzz led to the possibility for a television series South Park (1997), first at FOX and finally at Comedy Central.
When Santa spins Jesus around and throws Him away, Jesus lands next to a snowman wearing a hat. This is a reference to The Spirit of Christmas (1992), where Jesus fights a snowman which comes to live after the boys put a hat on it.