For the most part, Strangelove's mode of production was all about practicality. Example: it was filmed almost entirely in Shepperton Studios near London. Why England instead of the U.S.A? Because Peter Sellers was going through a nasty divorce and couldn't leave the U.K.
On another level, the demands of the script made doing the whole thing on a soundstage just make sense. Kubrick wanted a big creepy War Room, and he knew exactly how he wanted it to look, so he had to build it. The film also required the interior of a B-52, and, since real B-52's were high tech and top secret at the time, the filmmakers had to build their own version (which turned out to be very accurate).
Of course, even though most of the film crew was toiling away in London, a lucky (or unlucky) few had to fly over the Arctic to gather the footage of the icy scenery we see whizzing past Kong's B-52. The film was later cut up and painstakingly added in by hand. It might look a little fake now, but at the time the special effects were pretty sweet.
Why Black and White?
Lots of films were in color by the time Kubrick made Dr. Strangelove, so filming in black and white wasn't really necessary. Some claim that Kubrick chose B&W to give the whole movie the feel of black and white newsreels (click here for an example). This might be because he wanted to give the audience the feel that they were watching something that could really happen. Of course, some will tell you that this is a load of baloney, and really the only reason Strangelove is in black and white is because that was the only way for the film to stay on budget.
Regardless, the B&W film adds a note of drama and creepiness to the work. The last scene with Strangelove's monologue just wouldn't have been as effective in color; he looks like he belongs in Dr. Frankenstein's laboratory.
The film shifts back and forth between its three settings; all the action is going on at the same time but the people in the different settings don't know much of what's happening in the others. This was a huge editing challenge for Anthony Harvey, Kubrick's editor on the film. Here's his take on what happened:
One of the most remarkable things about the film is its deft cross-cutting between the three narratives unfolding in those locations in parallel time. Harvey recalls that getting all the pieces to fit wasn't at all easy:
"When the editor normally shows the first cut to the director, it never, never seems to be what you thought it might have been in the script. And in this particular case I remember we didn't think it was any good at all. So we just took the whole film to pieces, we had a huge kind of war room of our own in the cutting room, and we put up pieces of paper representing every sequence in different order. It took three months of working back and forth, thinking of the writer, did we get the right balance, you know?" (Source)
After all that work, one of the reels got lost and they had to start over from scratch. But did they get the right balance? Oh yeah.