What is the ruling regarding a woman who left her husband and married another man, cohabited with him once, then repented, and wishes to marry him lawfully after her divorce and completion of her 'iddah, knowing that her family might kill her if they find out? And is it permissible for her to marry without a guardian in this case?

The second marriage of a married woman is void, and she is considered an adulteress if she was aware of the unlawfulness of her action. You have done well by repenting, and there is no expiation required of her, but she must conceal her past. It is not permissible for her to seek divorce merely because she wishes to marry another man, but if she despises her husband, she has the right to seek khul’ (divorce at the wife's instance). We advise her to be patient, especially if she has children. If the khul’ is granted and her ’iddah (waiting period) expires, she may marry the other man if his condition is reformed and he has repented. It is not lawful for her to marry without the permission of her guardian.