> my fingers get cold when I wear rings

Unless the ring is shaped as a heatsink/radiator, I imagine it would eventually get into equilibrium, and you wouldn't feel the heat-flux.

Is it possible that the "coldness" comes from its indirect affect on blood-circulation?

Yes I think you are right and also fingers tend to swell and shrink with heat and cold so the ring that fits nicely in a cooler room will restrict circulation slightly once I'm feeling warmer, leaving me with one cold ring wearing finger.