"This woman hates us."
Translation:이 여자는 우리를 미워해요.
Why isnt sirohaeyo correct? It makes sense, and it's the only thing thats recommended by Duolingo
싫어하다 means to dislike. Hate is quite different. If duolingo is recommending 싫어하다 for hate then they need to change it.
합니다 is a more formal verb ending and you would use it like with maybe elders or I think they use it in the military. 해요 is more of a casual formal so you would use it with people who are older than you but not elders yet.