A woman will say "powinnam".
But of course here in our course every single sentence is said by a female voice, including sentences designed for men.
Why "Nie powinienem ciebie słuchać" here, but "Nie chcę cię słuchać" ["I don't want to listen to you"] in another exercise?
Are cię and ciebie interchangeable in these sentences?
Well, "ciebie" gives more emphasis... somehow it sounds pretty well to me here, although I guess "cię" would be better.
I'll answer both your half questions in this one post.
1) "I ought not to listen to you", though correct English, is IMHO rather awkward. Duo generally seems to favour "I should..." (rather than "I ought to...") to translate "powinienem" - as here: "I should not listen to you."
2) "I don't have to listen to you" is actually a translation of "Nie muszę ciebie słuchać" (as one_half_3544 and Jellei agreed, near the beginning of this topic), so it's incorrect here.
"don't need to" is a different thing.
"ought not" is rare, true. But it's just an accepted answer.
"should not" is the basic one.
Am I the only one that suffers a brain block with some sentences and make the same mistake over and over? This is one of them, time and again.