At the cabin, Cage tells a joke about a Buddhist monk ordering a hot dog. He actually ruins the joke when he messes up the punchline, saying "I'll have one with everything." However, the actual line is "Make me one with everything", a subtle but extremely important difference.
Although most of the film was shot in HD with Panavision Genesis cameras, about a week of shooting was done on 35mm film because the location was forecast to be over 110 degrees Fahrenheit, and the filmmakers were concerned that the relatively new and under-tested Genesis would overheat and fail.
Nicolas Cage came up with the idea for his character to be a Las Vegas magician, and for Jessica Biel's character to have a job that took her to a Native American reservation at The Grand Canyon. (information taken from the special features on the North American Blu-ray release)
(at around 42 mins) The Italian painter is Elio Carletti (1925-1980), not Carlotti. He was an Italian impressionist artist and an inspector for the Italian police. The Italian dubbing replaced him with the more famous Leon Battista Alberti who wrote a lot about beauty but, seemingly, not that quote.
Columbia Pictures was originally going to distribute the film and release it in September 2006 (it had a distribution deal with Revolution Studios), but they dropped the film in January 2007, and Paramount Pictures picked it up.