"No, we do not cook."
Translation:Нет, мы не готовим.
«нет» is a contraction of «не есть» meaning "there is not" but it is also used to mean "no" when on its own. So you'd see «нет» in places you'd otherwise see «есть» when the statement is negative.
(It's not quite true historically that it's short for «не есть» as it really came from older words «не е ту» which are no longer in use, but the meaning is the same so no harm no foul.)
I omitted "мы" and was marked wrong. Is "мы" necessary for this sentence construct?
I think it could depend on the question asked. But since there is no question, just translation, it seems logical that "мы" is needed.
Building on that, мы is "we" and "No, do not cook" is a completely different sentence.
We are готовим (present plural)
He is готовит (present masculine singular)