Does Islamic law permit divorce from an abusive husband who does not provide for his children, given that she does not possess the mahr to seek khulʿ? And is it possible to obtain custody of adult children (18 and 13 years old) if they refuse to live with him due to his ill-treatment?

If the situation is as described, then the wife has the right to seek divorce due to harm without forfeiting her rights, and she does not need to resort to Khul'. The custody of children after separation is a matter of scholarly disagreement regarding the age at which it ends. Some scholars hold that a daughter should be with her father after the age of seven, and a boy is given a choice. Others believe that both genders are given a choice after the age of seven. In case of a dispute over custody, it is necessary to refer the matter to the court to resolve the disagreement.