I think the related answer that James Kilginger provided is very good and worth a read. I'll touch on a few points related to your question.
There's 2 parts to this, how do ring systems form and where are they more likely to be stable and long lasting. I doubt that any ring system would be permanent, but some should be much longer lasting than others.
So, lets talk stability first.
As you pointed out, as ices melt inside the frost line (somewhere around Ceres, though different ices have different melting points, but in general, around Jupiter distance or greater ice particles are largely stable and can be part of a ring system (I'm not sure about Jupiter's gonzo magnetosphere though). Closer to the sun, certainly by the time you get to Mars, probably a bit further out than that, solar radiation melts ices and it turns into gas which the solar wind blows away from ring systems pretty quickly. What's more, any smaller rocky debris mixed with ice could experience spin or acceleration as ice is melted and escapes from one side as gas. Those local accelerations could blow some material from a ring system too.
Smaller dust particles in a ring system near a star would get blown away too by the Poynting-Robertson effect which is much more significant closer to a star. Some of these problems would be avoided if the star is in it's white dwarf stage and quite cold, but a main sequence star is much more friendly to distant ring systems than closer ones.
Ring systems are only stable inside the Roche Limit. That's covered in the other question and outside the Roche Limit, the material would tend to reform into a moon. Solid objects however can orbit inside a liquid or fine particle Roche limit and maintain their shape.
Planets close to the sun still have Roche limits usually well inside their Hill Spheres, but long term stability is still a problem for planets close to their stars. Here's an article that touches on this regarding Mercury and one on Venus. The same principal would apply to ring systems where the significantly larger solar tide would have a destabilizing effect on all objects in orbit around a planet close to it's star. It wouldn't destabilize immediately but over many orbits, objects could either spiral into the planet or outside the Roche limit, so Ring systems would be significantly shorter lived as planets get closer to their star. The tidal force grows with the square of the distance. Earth, for example experiences 90 times the tidal force from the Sun that Saturn experiences.
Saturn is also about 95 times the mass of the Earth, so it's not hard to see why Saturn is a much better candidate for a long term ring system than one of the inner planets. Much more mass and much further form the sun.
Formation of ring systems can happen by 3 methods, (that I can think of). Large collision, break-up of a moon that travels inside the Roche limit or accretion. The collision can be on a planet's surface where debris is blown into orbit or similar high speed impact on a moon in orbit around the planet. Collisions of this size are rare, but likely more common in young solar systems.
A moon that moves too close to a planet can get broken up and form into a ring system if it travels inside the Roche limit see pictures top right. There's no reason why that couldn't happen from time to time on a number of planets. It might one day happen to Phobos as it spirals in closer to Mars. Moons tend to move towards or away from planets quite slowly so this doesn't happen often, but it can happen from time to time, even when solar-systems are billions of years old and mostly settled into stable orbits.
Finally, accretion, Enceladus see here is feeding Saturn's e-ring and may have fed other rings in the past. Enceladus is unique among all the moons in our solar-system in that it's able to steadily feed a ring system. This is possible because it's quite small, so volcanoes on it's surface are able to eject material at greater than it's escape velocity and because it probably has a liquid water-like layer under it's surface, maintained by tidal heating because it's close enough to Saturn to experience enormous tidal forces. Enceladus is likely responsible for only a tiny part of Saturn's rings, but this is still a viable method for ring formation.
Europa on Jupiter experiences similar consistent volcanism due to large tidal forces that also keep some of it's inner layers liquid, but Europa's a much larger moon, so far less material gets ejected at or above escape velocity than with Enceladus. Only small moons close to large planets can supply material to a ring system in this way.
But, because moons may play a role in ring formation and Large planets are more likely to have more moons due to greater gravity and Planets far away from their star are also more likely to capture more moons because they have larger Hill Spheres and objects don't orbit the sun quite as fast, that's probably the gist of it.
Large planets are more likely to have moons and rings than small planets and further away from the sun, a planet is more likely to maintain a ring system then close to the sun. It's not impossible that planets close to a star or smaller planets could have a ring system, but quite a bit less likely.
A smaller planet means the rings would have to be closer to the
planet, which means the radial shear (difference in orbital speed vs.
radius) in the rings would be higher. A higher radial sheer would be
more likely to cause turbulence...or so I would think.
High orbital speed, I suspect, makes little difference, in fact, a larger planet would have higher orbital speed in it's rings than a smaller one. A heavy Jupiter, 8 or 10 Jupiter masses and 2 or 3 times denser than the Earth could have a large orbital region inside it's Roche limit and potentially, very large, very fast orbiting rings.
Hope that wasn't too long.