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 Nathan Fillion's pants split, hence the name.
When Mal and Kaylee enter the ballroom, Kaylee points out the chandelier. When the camera pans to it, a huge gold ring with elvish script can be seen in the center. It is the one ring from Lord of the Rings.
The part of recurring character Badger was originally written to be played by creator and executive producer Joss Whedon himself (according to Joss' DVD commentary for the Serenity (2002). In the end, actor Mark Sheppard was cast in the part instead.
Kaylee's full name is announced at the ball as Kaywinnit Lee Frye.
On Netflix, Hulu, and the DVD/Blu-Ray box sets this is episode 4, not 6. This is because they are in the order originally intended by Joss Whedon, not the broadcast order.