Looks like you just described a brownian motion and the associated diffusion process. As time goes by, this is still centered on zero but the variance increase with time.
Yes, any random process where you're accumulating the effects of a large number of random variables is the same in this regard. It's worth noting, however, that it is only the collapse of the cloud that makes the net angular momentum visible: without that collapse, the net angular momentum would be so small compared to the size of the cloud that it'd be completely imperceptible. Sure, the variance increases with the size of the cloud, but the size increases faster.
What do you mean by "correlated rotations"?
Well, I'll give a couple of examples. First, nearly all of the planets orbiting the Sun rotate in the same direction. I believe this was likely caused by shear in the protoplanetary disk that our solar system formed from.
Another example is with certain moons and Mercury: they rotate at just the right speed so that the same face always points towards the body they rotate. Our own moon is like this, as is Io. I'm sure there are others as well, but I don't know of any. This stems from the tidal forces: bodies in the solar system aren't perfectly spherical, and aren't perfectly solid. The slight deformations make it so that relative rotation of orbiting bodies is opposed. So, for example, as time goes on the Earth will slow down its rotation, as its kinetic energy is transferred to the Moon, which will achieve a higher orbit. This will continue until the Earth and Moon always show the same face to one another.
Or at least it would continue if our Sun didn't go red giant before that can happen, which it will.
Ok, here's my understanding of the flat things:
The flat stuff in the universe (some galaxy like ours, our solar system) seems to be made by something initially spherical getting smaller and smaller due to the effect of gravity. Being smaller means that it turns faster (like the second move of the girl here:
http://www.youtube.com/watch?v=g2v5iadTh9o)
So fast that at some point the centrifuge force becomes bigger than the gravity and shlouf, that nice little sphere gets spread out all over the place, but within the place perpendicular to the rotation axis (more or less).
Well, no. As Davros correctly points out, the centrifugal force is in all directions. It cannot cause objects that were orbiting off-axis to start rotating within a plane. And furthermore, the gas cloud doesn't start off as a sphere: it starts as an amorphous blob.
But as it collapses and gets more dense, it experiences friction. As it collapses one direction of rotation starts to dominate. Friction brings the rest of the galaxy in line, if the conditions are right (astrophysicists are still working on what those conditions are...what makes for a spiral galaxy versus an elliptical galaxy is still poorly understood).