Brandt,
I do not know what kind of silver solder you had but it is unlikely to contain gold in the alloy. By all means save and test it for values though.
As to cadmium, forget about testing for its presence for now (I know of no good method) but be aware of its hazards an how to eliminate it, know it goes into solution with nitric, and that silver cements on copper but cadmium will not.
Whatever you had in the bottom of your beaker that did not go into nitric now has some of your silver mixed with it as silver chloride. When you add a chloride salt (I imagine you used table salt) to a nitric solution that contains silver, lead, or mercury, you will form solids of these metals as their chlorides. What you should have done after digesting your silver solder is to remove the solids from your solution. Then you could have placed copper bar in your clean solution to cement your silver out, leaving any cadmium or other base metals behind in the solution. Careful washing of the cement silver would leave you with 98-99% pure silver.
If you wish fine silver you will have to learn about silver cells.
