1. Make sure you don't build a fire in the forest. Don't cook food in the forest as well.

2. Make sure you don't build a fire in the forest. Don't cook food in the forest,either.
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Q1: Is the sentence in #1 an affirmative sentence or a negative sentence?
Q2: Do we have to use 'as well' or 'either' in this case?

It's negative. Don't (do not) is negative.

#2 is better (either), but both one and two mean the same thing.