1) No
2) No. In order for the khul’ to be valid, it is stipulated that the woman hates the husband with a hatred that does not establish the limits of God with him and does not obey his command, so she gives him money to divorce her. If he accepts that and divorces her, it is fine, otherwise she cannot divorce herself.
3) The answer is no for both questions. She can divorce herself only if she stipulated in the marriage contract that she be his representative to divorce herself if he married another woman.