The Oort Cloud is continuous and its density falls off with distance, so it can’t be said to be at a specific distance. However there are estimates of an outer edge which place it around and occasionally beyond the Sun’s sphere of influence.

The Sun’s gravity is spherically symmetric, so it itself doesn’t exert any force to keep objects in a plane. Self-interaction of objects in the disk provides this force, through collisions or gravity. What the Sun does do is draw objects in, so the density of objects is greater nearer to the Sun, meaning there’s more self-interaction in the disk in these regions. The Oort cloud, though it likely formed in the disk, is far from the Sun and therefore underdense, so when orbits in the Hills cloud are perturbed and moved into the Oort cloud, there’s not enough self interaction to correct their inclination.

So, to answer your question: kinda, but maybe not in the way I interpreted your question to imply.

As the other commenter said, rotation of clouds is just the net effect of the orbits of their components. This is just speculation on my part, but I suspect the Hills cloud shows more rotation, because the objects that make it up are closer to their original orbits. It would also be rotating faster because it is more compact. In the outer cloud, I suspect there’s some small net rotation, but largely the velocities and orbits are randomly distributed so there’s less bulk movement.

The Sun formed from a collapsing molecular gas cloud, which itself had some net angular momentum. As the rotational velocity of this collapsing gas cloud increased due to conservation of angular momentum, it flattened out into the circumsolar disk.

It was in 1950 that Jan Oort first suggested that comets originated from a distant, sparse, spherical cloud. He proposed that comets were born in the circumsolar disk, in plane with the planets, before being perturbed by the gas giants and ejected into their spherical distribution. Subsequent studies of the orbits of comets and potential perturbations from passing stars strongly supported the existence of the Oort cloud, which was necessary because it is practically impossible to observe. J.G. Hills advanced the theory in 1981, suggesting that there exists an inner Oort cloud, now also dubbed the Hills cloud, that explains the mix of showers of short-axis comets and individual long-axis comets. The Hills cloud would be significantly denser than the Oort cloud, and works like this one used simulations to show first that it is indeed possible to get an Oort cloud from comets formed in the outer planetary region of the circumsolar disk due to combined perturbations by the galactic tide and gas giants, and second that the inclination of orbits only becomes random (spherical) beyond 5,000 AU due to the influence of passing stars and gas clouds.

So the much denser Hills cloud, which starts around 3,000 AU, is actually roughly torus-shaped like you expect of the Oort cloud. However the outer Oort cloud, which did likely start out in the orbital plane of the solar system, has probably been scattered by the combined effect of initial scattering by gas giants, and then further perturbation by extrasolar objects.