Translation:The upset man has closed the door.
What about "irritated"? It seems to me that "zdenerwowany" is a false friend and doesn't really mean "nervous". If I am nervous it means I'm apprehensive that something bad might happen. I'm trying to think of a Polish equivalent - "lękliwy" maybe? Zdenerwowany means more "wound up" doesn't it? That's how everyone I know seems to use it.
I would assume that trzasnął drzwiami focuses on the sound the door makes (if you can't see the door closing but you can hear it), whereas zatrzasnął drzwi is more visual. The language corpus suggests a usage ratio of 2:3 in favour of the instrumental, so it seems that both are quite common.