does this mean she is not acting responsible? or that she doesn't have the responsibility/ is not responsible for an accident/etc?
You're right, and it is exactly the same in English in fact. RESPONSible implies that you should respond to all (Dutch goes for answer "antwoord"). So in other words you are in charge/responsible for the situation.
I had the same feeling and I kept thinking of a word that gives the same meaning before giving up and checking tips :))
I think you're right, but I also think it's not always straight forward, e.g. Zij is niet erg verantwoordelijk (She is not very responsible) changes the meaning to not acting responsible. (Very big difference, compared to adding erg in a different sentence: Dit is niet (erg) hoog = This is not (very) high.)