Is it permissible for a woman to seek divorce from her husband who slandered her honor and humiliated her, and he refuses to divorce her and threatens to keep her in a suspended state? And is there any sin upon her for that or for leaving her children with him?

If the husband's slander of his wife's honor in front of her family is true, then he has committed a grave transgression that contradicts good companionship, and this is considered a violation of the Prophet's (PBUH) وصية (injunction/advice) regarding women. The wife is permitted to request a divorce in cases of severe harm, and the presence of children is not a Shari'ah impediment. However, one should deliberate before seeking a divorce, out of concern for the children's welfare. Custody is a right for the mother and is not obligatory upon her unless there is no other custodian available.