Yes this is correct. In -его/-ого endings, the "г" sounds like "в". Also in the word сегодня.
I know "ничего" is set earlier in the sentence to stress importance on "nothing", but can the sentence be constructed like this and still be correct: "Он не пьет ничего"?
Yes, it is different. Your sentence is translated as "Ему нечего пить". The original one implies that he has something to drink, but he doesn't drink. He refuses to drink.
I said he doesn't drink, which was wrong. How would one say he doesn't drink as opposed to he is not drinking anything?
It has a "don't want to drink" dimension, so it's not exactly the same meaning. Even if it's close, "He doesn't drink anything" is a better (and simpler) translation.
Your sentence would mean "he doesn't drink anything beer", so you have to change it. I think you can use "никогда" (never) instead of ничего, then it would make "Он никогда не пьёт пиво ; He never drinks (any) beer" I'm not sure, so if somebody sees it's wrong, please correct me !
It couldn't, because this double negation is the Russians' usual way to do it. You have to put both negations in order to have a correct sentence :p