Dua & Supplication

Is it permissible to make dua against someone who wronged you, and are you responsible for what happens to them?

Making dua against someone who has wronged you is completely permissible and is in fact one of the duas most guaranteed to be answered. The Prophet (ﷺ) explicitly said to beware of the dua of the oppressed (mazlum), as there is no barrier between it and Allah. You are not responsible for what happens to the person you made dua against — Allah decides outcomes, not you. The death of someone connected to the wrongdoer is not something you caused and requires no kaffarah. You should, however, only ask for what is commensurate with the wrong done to you.