Apurva- That's a complicated many-part question. I'll touch on a couple of points, then get to the core of it.
The gun cannot "fire it in one direction" because the aperture of the gun has some finite size. Any wave coming out of it has finite sideways spread, so that mathematically requires a spread of sideways wavelengths. That translates to a spread of sideways velocities. The relation between those position and velocity spreads is described by the uncertainty principle.
I'm not sure I understand the part about infinitely separated slits. For any finite separation the basic form of the math is the same. In practice, too much separation causes the interference to get scrambled up by outside influences. ("decoherence")
Now for the core. Forget about the electron being anything but a wave. Now all that interference part stops being mysterious. The part that might seem mysterious now occurs later. When the wave (from many electrons) hits a screen and you look at the screen, you don't see a vague steady glow, but little flashes of light, each from small spots on the screen. The spots are randomly located in regions where the square of the wavefunction is large. Why does this happen?
It turns out that if you try to write a wave equation for a bigger system, not just the electrons but also all the atoms in the screen and the molecules in your eyes, and include some term for all the distant objects which are slightly affected by the wave, you get something interesting. The wave breaks up into parts which no longer show interference with each other. They're said to have "decohered". One way to think of it is that each of the parts goes along with an outside world that has interacted with a different part of the wave. What sort of parts does the wave break up into? That depends on what the physical situation is, what sort of "measurement" apparatus the initial little electron wave ran into. If it ran into a tv screen, it decoheres into little blips at different spots.
So why don't you see all those blips at once, if they're all in the wave? Here's where things get a little controversial. If you're part of the wave, then your wave also breaks up into different decoherent parts, each "seeing" something a little different.
Is that the end of the story? Does every event end up with many slightly different versions of you (and the rest of the world) endlessly branching out? That's called the Many Worlds interpretation of quantum mechanics. Needless to say, not everyone accepts it as right! It has problems accounting for why the probabilities come out the way they do. And people just tend not to believe in all those other versions of themselves. One standard idea is that somehow the wave collapses, and only one version goes forward. There are big problems with that idea too. Various other more or less philosophical ideas have been put forward.
Fortunately, we can go on using quantum mechanics without figuring out that interpretation part.
(published on 12/24/10)