According to Nathan Fillion, he personally supplied the blueprints used to build the ship set in the film after being informed at a production meeting that the originals had been lost. He was able to do so because, being so excited about being in the original show, he had taken photos of ALL the preproduction materials he had seen, including the blueprints.
In the cargo bay, just after a Reaver is shot, some of the crates behind River have the message "Reusable Container: Do Not Destroy" printed on them. It's an inside joke. The ship set had to be rebuilt from scratch for the movie because the original, from Firefly (2002), was destroyed.
In his DVD commentary, Joss Whedon said Chiwetel Ejiofor, Nathan Fillion, Summer Glau, and Sean Maher performed many of their own stunts. Glau, a trained dancer, used her dancing skills in River's two fights. For the second fight, the cameraman, who was also a dancer, moved through a gap around the combatants to achieve the movie's dynamic camera angles.
According to the director commentary, Mal's line, "Faster would be better," was ad-libbed by Nathan Fillion when Joss Whedon told him to "say something Mal would say."
The cast had a running gag where they would yell Summer Glau's name whenever any of them flubbed a line or forgot to do something. It originated on Firefly (2002) when she forgot her line at the end of a very long and difficult scene. (Nathan Fillion, Morena Baccarin and Sean Maher can be seen doing this on the DVD blooper reel.)