When Kaylee calls Mal "Captain Tightpants" at the party, this was an inside joke referring to an incident that occurred while shooting the original pilot episode "Serenity". During the scene when Kaylee gets shot and Mal bends over to help her the back of his pants split, hence the name.
When Mal and Kaylee enter the ballroom, Kaylee points out the chandelier. When the camera pans to the chandelier, a huge gold ring with elvish script can be seen in the center of the chandelier. It is the one ring from Lord of the Rings.
Kaylee's full name is announced at the ball as Kaywinnit Lee Frye.
On Netflix and Hulu this is episode 4 not 6. This is because on Netflix they are in the order originally intended by Joss Whedon. The same order as the DVD box set.
The part of recurring character Badger was originally written to be played by creator and executive producer Joss Whedon himself (according to Joss's DVD commentary for the Serenity (2002). In the end, actor Mark Sheppard was cast in the part instead.