If the wife initiates divorce and ask for her husband's agreeance and he refuses saying that he would never give a divorce and to seek other means to do so, if the wife had a right to ask for a divorce due to being refused her rights and infidelity, first does he have the right to refuse, and second does it become a talaq because he refused to agree that day but eventually divorces a couple days later after he doesn't come home and she puts his things out saying she would send him the paperwork to finalize everything.? Wouldn't that still be considered a khula since she was forced to seek elsewhere because he wouldn't agree at first? If he only divorced her because she was putting his things out because he refused to divorce her, come home, and entertaining a woman that was not halal, is that still considered a divorce or an annulment?
بسم اللہ الرحمن الرحیم
Your question is vague, ambiguous and contradictory. Instead of ifs and buts, clearly write down the situation to find out the ruling in this case.
However, if the husband is unable to extend to his wife, her due rights, the wife can then demand khula or divorce from her husband.
However, the ruling on Khula or divorce can be given only after thoroughly examining the reasons behind it.