We don't know exactly how rings are generated by ED's Stellar Forge algorithms; it might go through the create moon > calculate Roche limits > destroy moons that are too close route, or it might simply accrete them out of the protoplanetary disc as part of the formation of the moon system around the planet. I strongly suspect that each discrete "ring" around a planet is actually a "moon", created but then destroyed to make the ring.
Ring type is very largely dependent on temperature. For Metallic rings, you're more likely to find them around planets orbiting the super-hot star classes (B and above). Likewise, if a planet has multiple rings, it is the innermost ones that are likeliest to be metallic.
As for ring formation, there seems to be a general "dustiness factor" for a star system, which affects the probability of ring generation; the more dust, the higher likelihood of rings. We've all seen those star systems where practically every planet and many of the moons have rings.
The star system's age also seems to play a factor, which makes sense; "new" star sytsems are going to still be full of gas, dust and debris to make rings from. Since big hot stars are always young, these two effects combine to make B and O class stars prime hunting grounds for metallic rings. Protostars (T Tauris and AeBe Herbigs) ought to be good for ring-hunting too, but are likelier to be the colder ring types rather than Metallic.
There also seems to be a high percentage of ring formation around "chaotic" planets in highly inclined or eccentric orbits. This is certainly true for moons, where unless the system is really "dusty", the only ringed moons are the outermost ones, the "captured rogues" orbiting at high inclinations to the rest of the satellite system. On the other hand, a stellar cataclysm (supernova explosion) seems to strip planets of their rings, as ringed worlds around neutron stars and black holes seem to be much scarcer than normal.
One final and very important factor in ring generation is the planet's gravity; high gravity planets are likelier to have rings, and are more likely to have more than one ring. As gas giants typically have very high gravity compared to dirtball planets, they have a high likelihood of generating rings. In my surveys, I don't keep stats on whether planets have rings or not, but just from "gut feeling" it seems gas giants have a greater than 50% chance of having rings. Earth-likes, on the other hand, only have about a 1.5% chance of having rings. This difference is entirely due to gravity.