I wouldn't say so. "Regularly" just means "at regular intervals"; it doesn't really say anything about how often. Regularly could be once a month, or once a year. "Frequently" means that it happens a lot.
It seems to me that the translation, without the word "the", implies that all men frequently drink beer, which I don't believe to be true. Wouldn't it be necessary to include "the" in the translation in order to be accurate?
Yes, long ago we gave up expecting the truth from Duolingo. However, it seems to me that the original statement, which includes "the" specifies a certain group. the translation, without "the" implies all men. Am I wrong in drawing this distinction when translating?
Well, you're quite right about the distinction in English. However, the French do not make the same distinction. "Les hommes" can mean "The men" or just "Men". I believe either is accepted in this exercise.
No. You can't use the English progressive tense with an adverb like "frequently". "Are drinking" means it's happening right now. "Frequently" refers to something that happens over and over again. They don't go together.