Acting for Two: Peter Sellers as Captain Mandrake, President Merkin Muffley and Dr. Strangelove, and it was meant to be 4 but a sprained ankle prevented him from getting into and out of the B-52 set, so Slim Pickens was added to the cast to play "King" Kong.
Dueling Movies: Fail Safe, a dead serious take on this Failsafe Failure premise, was also released in 1964. The straight film is good (though it performed poorly at the box-office), but Kubrick's film has become iconic.
Slim Pickens was unaware that the film was a comedy. This is possibly because previous actors, including John Wayne, had turned the part down because they saw the film as "pinko." That said, Pickens didn't have any problem with the film, especially since he became so famous for it, though this may be because Kubrick only gave him his parts of the script.
Also, George C Scott was unwilling to go over the top in his portrayal of General Turgidson, so Kubrick tricked him by telling him to do a few over the top takes as "practice" and that they would never be put into the real movie. Kubrick lied, creating one of the best Large Hams ever but also causing Scott to swear he'd never work with Kubrick again. Scott did end up admiring Kubrick's genius behind all that deception.
Executive Meddling: The geniuses at Columbia Pictures were for some reason under the impression that the only reason Lolita was a success was the gimmick of Peter Sellers playing multiple roles. They would only greenlight Dr. Strangelove on the condition that Kubrick agree to cast Sellers in at least four roles.
Fake Nationality - British-born Peter Sellers plays the American President and the eponymous German scientist, as well as the British Group Captain.
Much of Peter Sellers' dialogue was improvised (Kubrick had three cameras on Sellers at all times to take full advantage of this), including the hotline telephone conversation; similarly, Strangelove's Evil Hand's rampage at the end was largely improvisation (it was also Sellers' idea that it should be gloved). Keep in mind that Kubrick was the most insane perfectionist in the history of filmmaking. And Sellers got to ad-lib.
At one point, George C. Scott is emphatically trying to convince the president (Sellers) to launch an attack when he suddenly trips and collapses to the ground before quickly picking himself up and continuing the scene. It fit so well with the deliberate ridiculousness of the rest of the movie that Kubrick left it in.
Too Soon: The release date was slightly delayed after JFK's assassination due to the story involving a fictional president:
In the scene where Major Kong reads the description of a survival kit's contents out loud, he originally says "Shoot, a fella could have a pretty good weekend in Dallas with all that stuff!" Dallas was redubbed to Vegas because of the connotations carried by Dallas in a post JFK-assassination America. It actually worked out well, as Vegas makes more sense in this context, especially today — no one would think of Dallas as a "party destination" any more, but Vegas is still the king of "What happens here stays here".
The original ending was to have everyone in the war room end up in a pie fight (don't ask). The President would be knocked down from the impact of the pie hitting him, with Gen. Turgidson saying "Gentlemen! Our gallant young president has been struck down in his prime!" Despite it being filmed before the assassination...wow. Just wow. (That wasn't why the scene was deleted, though- they just couldn't film it with the necessary "gravity.")
What Could Have Been: Peter Sellers originally was also asked to play Maj. T.J. "King" Kong, and practiced intensely with the American screenwriter to get the American Accent right. After the first day of shooting, he sprained his ankle, and could no longer work in the cramped airplane set. So they recast the role with Slim Pickens.
John Wayne was considered for the part of Major Kong as well.