"We are not wearing a hat." would be highly unusual in English as it would not mean each. It would mean there was a very large hat that we are not wearing. We would have to say "We are not each wearing a hat." or "We are not wearing hats." Of course, to be safe, I put "We wear no hat." which was accepted, but it should have been "We wear no hats."
In the end of the lesson there is a task with this sentence: 'Dieren dragen geen klompen'. Plural 'klompen' is forced there as the only correct answer among two possible options: 'klomp' and 'klompen'. Why plural 'Wij' + singular 'houd' is allowed here and plural 'Dieren' and singular 'klomp' are not allowed there? Seems like an inconsistency.
"Wij dragen hoeden," simply means, "We're wearing hats," or "We wear hats." Although I believe you can interpret it as each person wearing multiple hats, it usually wouldn't be seen that way. "We're wearing hats," works just fine for each person wearing one hat. :)
Once again, "Wij dragen een hoed," could probably be seen as one hat being worn on all of them, but it's pretty much always going to be seen as, "We are wearing a hat," as in they are each wearing one hat.
At least that's what I think. I'm not an expert, but I'd be surprised if that wasn't the case.
I'm a native English speaker, but I'm not sure why your translation wouldn't work. Perhaps it is too general, in that you are saying we don't wear any type of hat, when this exercise is looking for something meaning, "we are not wearing (a) hat(s) right now," specifically?
Oh, whoops. Apparently I didn't look at that very carefully. XD Well, I'm not entirely sure what you mean. As long as there isn't a definite article - de or het - or a possessive pronoun, you use "geen" instead of "niet." Like I said, if there is no article it isn't indefinite. It just uses the same word when negating. Here are some examples:
"Wij dragen geen hoed." = "We are not wearing a hat." In this instance an indefinite article is used, so you use "geen."
"Wij dragen geen hoeden." = "We are not wearing hats." In this instance there is no article - it is not indefinite. However, you still use "geen," as there is no definite article.
"Wij dragen de hoeden niet." = "We do not wear the hats." This does use a definite article, so "niet" is used.
"Wij dragen onze hoeden niet." = We do not wear our hats." In this instance, a possessive pronoun is being used, so "niet" is used.
Hope this answers your question. :)