In our solar system, with the exception of Pluto all planets follow a relatively circular orbit around the Sun, at the same inclination. They also all rotate in the same direction, none are 'retrograde'.
An image of the orbits of our solar system.
How and why have they all formed into these orbits? Is this reflected in most orbital systems? What factors affect the orbits of newly formed systems?
The combination of the conservation of angular momentum and gravity give you the inclination and direction. Gravity will condense material in the axis direction of the angular momentum, and collisions and gravitational interactions will damp oscillations in that direction, forming a disk of material. The angular momentum resists gravity in the plane perpendicular to the axis, maintaining the extent of the disk. Over time, the direction of rotation of the angular momentum is the direction of the vast majority of the material, due to collisions and interactions with that smaller population going the wrong way on a one-way street.
The circularity of the orbits is the result of a more dynamical process. Hopefully someone else here can explain it better than the following. My simplistic understanding is that a bunch of orbits crossing each other is not stable. Planets and planetesimals have their orbits changed continuously until such time as they settle into orbits that have fewer interactions with other bodies. Sort of a natural selection. In the long haul, this results in a relatively stable configuration with circular zones that tend to not interfere with the other circular zones. In fact, this is now part of the definition of the word "planet", in that to be called such, a planet needs to clear the neighborhood of its orbit. Though its not clear how it could occur, a bunch of elliptical zones that happen to be co-aligned would also not be stable since orbits precess due to various influences.
Where the material was less dense than where the planets are currently must certainly have many objects in eccentric, retrograde, and/or highly inclined orbits. Pluto is the first hint at that, with a relatively high inclination and eccentricity compared to the planets. Sedna is far more eccentric.
As for other systems, I'm sure there must be some oddballs out there. However the Kepler mission has observed mostly circular orbits for planets around other stars.
The rings of Saturn are only about 10 meters thick. What phenomenon pulls the particles, over time, into a thin disk?
Consider a particle that is in a circular orbit inside the ring. Its velocity can be resolved into an in-disk component, Vd, and a component, Vn, that is normal to the disk. Since the orbit is circular, the radial component of velocity is zero. Non-zero Vn will cause the particle to rise out of the plane of the ring. But when it does, the gravity of the other particles will pull it back. This will change the sign of Vn and cause the particle to fall back toward the ring plane. Thus Vn will oscillate over time while Vd remains constant, and the particle will continue to swing above and below the ring plane until it collides with another particle.
After the collision, each of the particles involved with emerge with new values of Vd and Vn. If one of them has Vn near zero, then it becomes a part of the thin ring and unlikely to suffer future collisions as it now "goes with the flow." If the other particle has non-zero Vn, it will continue to oscillate and face the prospect of further collisions. Over time, more and more of the particles will survive their collisions with Vn near zero and thereby become "good citizens" of the ring. The particles bouncing out with substantial Vn will continue to be battered about until they too settle down and "get in line." This argument applies to a developing thick ring as well as a developed thin ring.
