Yes, you are right. 別喝酒

That would be 我不喝酒

It just depends on context really, so technically yes. If need be, you could say 我不要喝酒。

You'll usually still need the subject for that to make sense: 我不要喝酒 (I don't want to drink alcohol) or 我不喝酒 (I don't drink alcohol). Otherwise, like English, it's implied that you are talking about someone else like a command rather than about yourself.

I also think so, I used sentence like this when i was in china as "i dont (want) to drink alocohol"

Or would you always have to put 我 to the top of the sentence?

酒 means alcohol. 啤酒 means beer. The terms that Duo uses for white wine and red wine (白酒 and 红酒) are criticized by many people in the forums, (including me, since when I was in China, 白酒 meant a Chinese-made, clear-colored, hard alcohol!)