Why do you suppose the rings are essentially in two dimensions (a disk) rather than spherical?
Because they are revolving around the proto-sun. As the Brownian motion of the particles making up the interstellar cloud is organized, they begin to rotate. This is the same reason the water in a flushed toilet rotates.
By convention, solar systems and planets rotate counter-clockwise (there are exceptions). The galaxy rotates clockwise. This has to do with the conservation of angular momentum.
IIRC - The rings of Saturn are only about 30m thick. The thickness of the asteroid belt is harder to nail down as their orbits vary greatly. Asteroids are spread very thinly. Half the mass of the asteroid belt is contained in just four asteroids.
Hope this helps.
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LonePalm, le Républicain du verre cassé (The Broken Glass Republican)
Gravity well...........
Conservation of angular momentum