(at around 4 mins) The "girl with necklace" at his magic show is played by Alice Kim Cage, Nicolas Cage's wife.
The diner where Nicolas Cage runs into Jessica Biel's character is the same diner used in xXx (2002) with Vin Diesel.
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.
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.
Throughout most of the film, Cage wears a gold jacket. This is a nod to the original Philip K. Dick story "The Golden Man" upon which the film is based.
The Cliffhanger motel is actually a restaurant located on Highway 18 near Crestline, CA, in the San Bernardino Mountains.
(at around 53 mins) When Nicolas Cage is zapping through TV channels, he stops at one showing the movie Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb (1964), leaving it on for the FBI to listen to. The plot of "Dr. Strangelove" is about a rogue general attempting to start a nuclear holocaust with a wayward nuke. An obvious reference to the nuke in this film.
(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.
All three helicopters (2 black, one white) are Eurocopter (nee Aerospatiale) AS-350 Ecureuils (Squirrels). The N-numbers were all sequential as well - N350SC, N351SC, and N352SC.
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)