Hal Schaefer wrote a score for the film but director Sidney Lumet decided to release it without any music. However a promotional single by The Hal Schaefer Quintet, "Fail Safe Parts 1 and 2" was released on the Colpix label. There is music on the trailer, and it is probably extracted from Schaefer's score.
The view of the satellite zooming into a closer shot is actually film taken from a camera mounted on a captured German V2 rocket launched from White Sands, New Mexico. The film is run backwards to show the illusion of zooming closer to the ground.
With one exception (see goofs entry), all shots of US Air Force "Vindicator" bombers are views of the same Convair B-58 Hustler, taken from a stock piece of film after the Department of Defense declined to cooperate with the filmmakers.
The big screen in the control room at Omaha was entirely front-projection, and had to be very carefully contrasted to appear clearly on black and white film. This posed a problem for the crew, as the air in the room had to be totally clear of dust so as not to disrupt the image (and make the projection obvious). The screen in the war-room used the same film-stock but was rear-projection.
Some reference works (eg Colonna Sonora) credit Hal Schaefer as composer of the music score. In fact, the Columbia record division did issue his five minute Fail Safe suite on a 7-inch disc in 1964 (Colpix Records CP 751) with the film promoted on the actual label: FAIL SAFE - A Columbia Pictures Release. Schaefer's music, performed by the Hal Schaefer Quintet, was a somewhat nervous and dissonant jazz work, fairly appropriate to the film's content. However, director Lumet decided to release the film with no background score, so Schaefer's music only exists on the promotional disc.
In the original novel, the character of Dr. Groeteschele, played by Walter Matthau, has a back story very similar to the early life of future Secretary of State Henry Kissinger: a German Jewish immigrant who fled Nazi Germany with his parents as a teenager, then became an intelligence officer in the US Army in World War II in charge of interrogating German Prisoners of War, then went on to become a specialist in International Relations.
Although the first view of Henry Fonda is a full-body shot of him walking down the hallway, his first closeup is from the back, as he looks over his shoulder with a smile. This is similar to the surprise view of Fonda as the bad guy in Once Upon a Time in the West, showing his face for the first time as a young boy is killed.