Keep your silver jewelry in the small ziplocks to keep them from tarnishing. Works like a charm, especially if you squeeze out as much air as possible before sealing. Bonus - orderly storage, and you can see what's inside.
You're making a cup that you will presumably be drinking from. Lead from the solder can leach into the liquid you are drinking. That is why you do NOT want to use solder that contains lead.
Liver of Sulphur works very well on copper. You can get a nice gunmetal finish by dipping an object until it becomes dark (multiple dips are better than one long one, which might cause the patina to flake off), then rinsing and brushing with a brass brush. On plated surfaces, though, you run the risk of brushing through the thin layer of plating to the original metal surface. If you use a chemical or heat patina, be sure to seal the result with wax (Renaissance Wax is what I use) or a spray-on lacquer (matte finish is best, to avoid an artificially shiny result).