Do planetary rings have geometric bounds? Are there any bounds on where a planetary ring can form and maintain orbit relative to the planet?

Do they have to be in a circular equatorial orbit?

Is there a min or max altitude, say with respect to the Roche limit, geosynchronous orbit, or other points in space?
The Saturn's Phoebe ring that has an orbital inclination 173° to the ecliptic so it is actually in a retrograde orbit and is tilted 27° to Saturn's inner rings, shows that there clearly isn't any limit to orbital inclination. It is feeding off the Saturn's moon Phoebe (possibly due to micrometeorite impacts) indicating that as long as the planetary ring has a source of its materials with already inclined orbit, the ring will follow it (preserve the angular momentum of its originating body). That of course doesn't exclude polar orbits, but we have yet to see any rings like that. Theoretically though, their orbit is just as stable as any other as long as it doesn't intersect other gravitational disturbances, like Lagrange points, or paths of other celestials that would stop it forming a complete ring with accretion (something that is only partially done by Iapetus intersecting the Phoebe ring). Majority of planetary rings might be made of planet's own materials though, and would then naturally follow its own rotation, more or less (depending on their formation) and again preserving angular momentum.

Phebe ring is also extremely huge, stretching from calculated 59 to 300 Saturn's radii (observed by NASA's Spitzer Space Telescope in the infrared range between 128 and 207 Saturn's radii, more can be read in this Emily Lakdawalla's blog post), attesting that as long as the ring's materials would not reach escape velocity or be gravitationally attracted more by another celestial body (beyond the L1 point) there isn't much of a limit in their size either. In theory, if we for example imagine some rogue brown dwarf that isn't gravitationally bound to any solar system and floats freely in the interstellar medium not close to any stars, the ring size would only be limited by the interstellar medium's own pressure. At least until the brown dwarf in questions roams closer to some stronger gravitational influence and loses its ring, that is.

So the maximum size of the ring would in theory indeed be limited only to the L1 point (where gravitational attractions of two massive bodies cancel each other). As for the minimum distance, that depends on the size of particles that the ring is made of, stretch of planet's atmosphere, and strength of its radiation pressure that would be negating gravitational attraction. So it is difficult to give any fair minimum value. If we take a fast spinning planet (e.g. dwarf planet Haumea) as an example, one that would spin so fast that it would actually lose some of its materials due to it reaching escape velocity at the planet's equator, and if we assume this process can be sustained for long enough period, then there isn't any minimum as the planetary disk would be essentially touching the planet's surface. Eventually, the planet would lose much of its own rotation, similar like a fast spinning figure skater would by stretching her arms, by shifting some of its mass into larger radius, and the planet would cease replenishing disk's materials.