How long does this process take?
We don't know. As I discussed here1, Saturn's rings could have formed billions of years ago or only a few hundred million years ago. Wikipedia and NASA each explain the two different sides; the other links in my answer are also informative2. I like the 4-billion-years theory because there would have been a lot of extra (i.e. non-planet/comet/asteroid/whatever-forming) material in the original protoplanetary disk that needed to go somewhere. But I'm by no means an expert. Same goes for the other ringed planets (Jupiter, Uranus and Neptune), by the way.
Could these moons retain their non-equatorial orbits and, if so, could they influence planetary detritus in such a way to form rings in more than one planar space?
There's a reason why the planets (and, by extension, their moons) are all in the same plane.3 There are, however, some reasons for an orbit to not be in the same plane.4 The idea of a captured moon is the most probable here.
I don't trust reddit (no offense to all who use it) when it comes to science, but a few comments later down were rather interesting (emphasis mine):
Das_Mime: Planets bulge at the equator and thus will exert a tidal force that preferentially pulls objects into orbit around the equator by disrupting non-equatorial orbits.
NW: Will our moon eventually get pulled into an orbit around the equator?
Das_Mime: Its orbit is thought to be gradually moving closer to the equator over long time periods, but the closer it gets to the equator the weaker the effect is. When a body has multiple moons, they have a mutual gravitational interaction which can force the moons to orbit in the same plane or else be ejected, since that's the only stable configuration. Earth's Moon is a lone satellite and orbits at a somewhat large radius from the Earth (though not even close to as large an orbit as the outermost satellites of Saturn or Jupiter), and moreover the Earth's equatorial bulge is quite modest, and so the effect is rather weak.
Since Earth is not an especially oblate spheroid (a difference of about 0.3% between the polar radius and equatorial radius), its equatorial bulge doesn't have a terribly strong effect on the Moon's orbit. However, the tidal interaction between the Moon and the Earth's bulge (as well as between the Sun & the Earth's bulge) does cause the precession of the equinoxes, a change in the orientation of Earth's polar axis.
Saturn, on the other hand, has a fairly extreme oblateness, about a 10% difference between the polar radius and equatorial radius (it's even noticeable to the eye in images of the planet and so it has a much more pronounced effect on the orbital orientation of objects near it. Additionally, if we're talking specifically about Saturn's rings, they extend no more than ~150,000 km from Saturn's center (~60,000 km of which is taken up by Saturn), whereas Earth's Moon orbits at about 380,000 km from Earth's center (Earth is only ~6300 km in radius).
Another possibility is to have rings of Hot Jupiters warped by tidal forces from their parent star (see implications for detection in Tusnski & Valio (2011)). This was also mentioned in an answer by JRover, citing Wikipedia and Schlichting & Chan (2011).
2More shameless self-promotion.
3Man, I'm adding lots of plugs for Astronomy.
4I'll stop eventually.