That site looks really good, but most of the sites, including this one have a try before you buy option. I would choose a few, send them the same pictures and see what they come up with.
In regard to your setup:
Black does enhance the metal reflections, but I´d rather use black cardboard than cloth. How you setup your lighting depends a lot on how much time you have for every single picture.
You can use different sized pieces of black cardboard (or cut pieces of black painted styrofoam board which have the advantage that they stand on their own), and place them in a way that reflections are enhanced. English is not my main language, so I hope you understand what I mean. The blacks make the outlines, the contours of your rings for example. But placing them perfectly for every single piece of jewelry takes up quite some time. But you can also just leave them in place if you don´t want the perfect reflections, but just a few here and there.
Everything else has to be white. So a shooting tent with enough room to place your black pieces would be a good option.
If you can easily remove one side of the shooting tent to let the sun in (without moving the jewelry), that would be the perfect setup.
Then you´d shoot one image with sunlight (slightly darker camera settings - for the stones) and one image with closed sides for the rest. In your editing program you open these two in layers, roughtly mask the stones on the sunimage, and put it over the other. Then you flatten the image and send it to an editing service. That should give you the best possible quality in a rather quick workflow.