I actually found "organic" garlic at my walmart. I know what you're talking about with the Chinese garlic, it looks strange because it's almost too clean looking. The one at walmart is from Argentina. In my experience, some of the best garlic sold in the US came from Mexico. I say grow it yourself if you can, go with the Argentina or US product.

Don't cut anything out, and if it sprouted you can always plant or just cut it off. As long as it isn't rotting, you're good to go.

Garlic on an empty stomach is most likely more effective, make sure to chew it up real good so it's basically liquid going down. It's going to burn like hell, I know well. And take it from me, spread it out throughout the day, because eating 10+ cloves during one session is going to give you a garlic high and then the eventual coming down(at least in my experience).