"We do not understand why he does not close the door."
Translation:Nie rozumiemy, dlaczego on nie zamyka drzwi.
Using present simple indicates repetitive or habitual action, so zamyka is correct.
Also zamknie is future tense, if you used it the whole sentence would have completely different meaning.
Not every verb in Polish has dokonany (done) and niedokonany (in progress, repetitive) aspect in present tense. Zamykać is one of them.
It's hard to guess intentions at times. I came across this sentence as I was "testing out". I have not gone through the course, and much like in other languages there's a wiggle room, not always recognized by the program. One can imagine a situation to which this sentence pertains but the intent of the inventor may be different. Are you Polish?
Neither 'drzwa' nor 'drzwu' are existing words ;)
Don't focus too much on the 'negative' thing. Generally, different verbs and prepositions just need different things. True, there are a lot of somehow 'negative' verbs that need Genitive, but it doesn't guarantee it. And for sure there isn't anything like neg x neg = pos.
It actually is Genitive. It just happens to be identical to Accusative and Nominative with this specific word. So you will have "dlaczego on zamyka drzwi" and "dlaczego on nie zamyka drzwi", they use identical words, but these are different cases actually. Compare with closing 'a window': On zamyka okno vs On nie zamyka okna.