This marks the last time that Buffy's birthday is celebrated in the series. Spike comments that, because supernatural events have a history of occurring on Buffy's birthday, she should probably stop celebrating them. Buffy traditionally celebrates her birthday around the 12th or 13th episode of each season; her birthday takes place in episode 13 of Season Two (Innocence (1998)), episode 12 of Season Three (Helpless (1999)), episode 12 of Season Four (A New Man (2000)), and episode 13 of Season Five (Blood Ties (2001)).
The character of Richard is a classic Star Trek reference: A fresh-faced, never-before-seen character who, upon encountering a form of otherworldly life, is immediately injured, almost fatally. Add to this the fact that he's dressed in a red shirt, and you have an example of an early Star Trek (1966) throwaway character.
In Dawn's classroom, the teacher is saying, "... if we can come up with things Jim lost during his years in Shanghai, and things he gained." She is referring to author J.G. Ballard, whose autobiographical novel "Empire of the Sun" was adapted to film in Empire of the Sun (1987).
Xander makes reference to Children of the Corn (1984). Nicholas Brendon played a role in Children of the Corn III: Urban Harvest (1995).
When Dawn goes to her bedroom and everyone follows, she shouts "Get out. Get out! Get out! Get out!" She did the exact same thing a year before in Blood Ties (2001) to Joyce and Buffy. Coincidentally it was also during Buffy's birthday party both times.