"I am not drinking beer."
This is not right as Nepiju pivo would be I dont drink beer. I am not drinking beer refers to current action I am doing so it would make sense to use "ted" or "nyní" or "právě" to imply the current action occuring.
You are not correct. Nepiju pivo is a completely cromulent translation. No additional adverb is needed. In Czech some senteces are ambiguous when it comes to translating to English progressive or English simple tenses.