From what I learned in art school (so take with a grain of salt) your brain will compensate for the 3d effect for a period of time (so that you think that you're still seeing in 3-d) but it can't compensate fully for 3d, which means if someone throws a ball to you, it's really difficult to catch it.
Apparently there was an artist who wore an eyepatch for a few months, after which his brain stopped the 3d effect, and he essentially was seeing in 2d.
This of course was all stuff told to me in an art class on perspective, so take with a grain of salt.
Two eyes are required for ordinary depth perception without workarounds, but depth perception is not required for 3D vision.
Speaking as someone who naturally has little to no depth perception (amblyopia affecting my right eye), I still see 3D just fine, catch flying objects (balls, frisbees, etc.) just fine, and am not disabled in any significant way except an inability to see into those magic eye puzzles.
Two eyes give you the ability to detect parallax without moving your head; those of us without binocular vision detect parallax by slight (imperceptible) movements of our head. There are many other useful visual artifacts of our 3D world: closer things are larger than farther things, closer things obscure farther things, and so on. Our brains just construct 3D worlds based on those data rather than the simpler, more reliable stationary parallax detection that you two-eyed folks use.
People also use depth of field / focal blur to aid depth perception. This is a major reason why the brain has issues with fake 3d using separate images, you want to focus on a part of the image, but things get blurry if you don't focus on the screen.
Motion parallax is the name for that effect. It is definitely a clue in perceiving 3 dimensions. Relative size is another clue: if you know roughly how big something is and it looks smaller, you can assume it's far away. Level of detail, interfering haze, etc all are other cues.
But the difference between what each eye sees is also an important factor in 3D vision. Movies just make it the only factor, which is probably why they can be disorienting. My guess would be that it's more important for things that are close than things that are far away; something 3 inches from your face appears to be in drastically different positions if you cover alternate eyes, but distant mountains look about the same.
One very surprising example of someone playing with single eye vision. Mansur Ali Khan Pataudi (http://en.wikipedia.org/wiki/Mansoor_Ali_Khan_Pataudi) played in the Indian cricket team as a batter for many years. He lost one eye at the age of 20, but surprisingly still managed to bat.
This is particularly striking since batting involves judging the ball that comes at you with some speed. His previous judgement and muscle memory might be useful of course. But I'd love to read about any research on this, if anyone is aware...
Apparently one of the guys involved in early Air Force space suit research had lost an eye in an accident (no, not Yeager, can't remember the name right off). It created a bit of a stir when they tested him post-accident and discovered that his depth perception had actually /improved/ now that he only had one eye.
Unfortunately, despite this, they weren't willing to let him go back to flying fighter planes, which is why he got into space suit research...
Uninformed guess: the speed of a top-level cricket delivery (~100mph) is such that stereoscopic vision doesn't really play a role. It's all in the timing.
Squirrels have their eyes on the sides of their heads (herbivore norm). They live in trees, jumping from branch to branch.
They do it by moving their head - watch for a while, they almost always run a little bit along the branch before jumping. Sort of like synthetic aperture radar.
Apparently there was an artist who wore an eyepatch for a few months, after which his brain stopped the 3d effect, and he essentially was seeing in 2d.
This of course was all stuff told to me in an art class on perspective, so take with a grain of salt.