Hi :) For the backs I've used simple rubbing alcohol on a Q-tip (mostly to kill bacteria, but it cleans everything off, too). If you're getting a green color on the posts or backs over time, there's probably brass underneath plating that's reacting to your skin. Some people react to silver with a black "smudge", and some even to gold the same way. Sometimes chemicals you're around daily will cause a similar reaction. I've noticed painting with latex paint will cause my rings to leave black marks on my fingers.
Quick fix if you're reacting to earrings: clean off the smudge, gently wash the jewelry clean, then apply a layer of clear nail polish to the post and back. Works great, and when it eventually chips off, just put a new layer on :)
The site you linked to has some good information, but be very careful with baking soda and toothpaste - both are very abrasive and can damage metal, glass, and definitely anything with nacre on it. Lemon juice is also a bad idea and can react to pearls and shells (think acid). When I'm done working a silver or gold piece, I quench it in a strong acid, then rinse in a base like baking soda to neutralize it. Using only one or the other is risky, especially if you don't know what the reactions to various other materials are (like paua, soft stones like jade or lapis, and on and on). There's so much science behind it all that is fascinating - that's partly why I got started a long time ago!