I think "Lui non beve caffè nero" means he doesn't drink black coffee in general, whereas "lui beve il caffè nero" might might he does not drink a particular cup of black coffee (a better translation might be he isn't drinking the black coffee). I'm not Italian, but I know Spanish works this way, and, from what I've gathered here from these exercises, so does Italian.

No, because that would be "Lui non beve il suo caffe nero"

From what I understand, adjectives have to match their noun in number and gender. Also, masculine nouns can end in either -e or -o. "Caffe" and "nero" are both masculine and singular so even if they don't end in the same letter the sentence "Il caffe nero" is correct.