"You answer to me, not to her."
Translation:Tu îmi răspunzi mie, nu ei.
You use the stressed form in the second sentence because you want to emphasise the fact that you are answering me or should answer me and not her. The simple sentence would be: ”Îmi răspunzi” - You answer me. Plain and simple, no more information added. When you say: ”Îmi răspunzi mie”, you're stressing the fact that you should answer ME, not anyone else. Thus, following the same logical reasoning you also have to stress the person you ought not answer: ”Îmi răspunzi MIE, nu EI” - You answer ME, not HER / You give an answer to ME, not to HER. Hope it helped ;)
"I thought if I use îmi, I do not have to use mie as well"
That's usually the case. I think in this sentence it would be weird not to use the stressed pronoun (mie), though, because you're emphasizing the contrast between me (mie) and her (ei).
I'm still learning, so if I'm wrong I hope someone will correct me.