If space was n>3 dimensional, would clouds of particles still turn into 2D disks due to gravity? I don't fully understand why clouds of particles which turn into galaxies and solar systems usually result in an (approximately) 2D shape, but whatever. Would they still result in an almost 2D shape if our space was n dimensional with $ n &gt; 3 $ and the laws of physics were the same (adapted for that $ n $)? Or would it be an (approximately) $ n-1 $ dimensional shape?

It would be of interest to have an answer to this question under the assumption that the attractive force was still proportional to $1/r^2$ even when the dimensionality is greater than 3.
It just so happens that this problem has been rigorously analyzed.  Max Tegmark wrote an interesting paper on it which can be gotten from the arXiv.  It is not super-technical, so I recommend you look at it.  Basically, with more than three space dimensions there are no stable orbits. Particles either disperse to infinity or collapse to a point (presumably a black hole).  (Tegmark also analyzes the effect of extra time dimensions -- weird.)

Added: A comment asks if a cloud of particles behaves differently and might still be stable. The answer is "no".  Here's why.

There are two cases here "gas" and "dust". The difference is that gasses have internal pressure which does affect the cloud's dynamics, while the particles in dust are big enough that collisions between them become infrequent and can be ignored for their effect on the dynamics. (Note: the only important difference between "gas" and "dust" is if the particles interact often enough to produce enough pressure to significantly affect dynamics.  It's not their composition. The terms "gas" and "dust" are simply terms of art used to describe the two cases.)

The dust case is trivial: The dust particles spend nearly all of the time in their individual orbits without colliding and if the orbits are not stable, the dust cloud isn't either.

Gas is more complicated, but it's probably easiest to look at what happens in three dimensions.  A non-rotating gas cloud collapses -- that's how galaxies and stars form!  A gas cloud that's rotating enough to support itself against collapse is relying on the centrifugal force to support it against gravity -- which doesn't work in four or more dimensions.