Is it permissible for a divorced woman, whose father refused to marry her off—despite the man and his family's approval—to marry without his consent? And is it permissible for her maternal half-brother to be her guardian (wali)?

A guardian does not have the right to permanently prevent a woman from marrying. If a suitable suitor proposes to her, he is not permitted to prevent her marriage without a legitimate شرعي reason. Otherwise, he would be considered an oppressor (عاضل), and his guardianship would be revoked, transferring to the next closest relative or to the judge. The brother from the mother's side (أخ لأم) cannot be a woman's guardian in marriage because he is not an 'asabah (agnate). It is best to refer the matter to the شرعي judge, who will compel the closest guardian to marry her off, or order the next closest relative, or undertake her marriage himself.