What are the legitimate rights of the wife and children in the event of a divorce due to the wife's desire to marry another man, and is it permissible for her to marry her colleague after the divorce?

The husband erred by violating Islamic law, failing in his duty of qawamah (guardianship/financial support). The wife erred by being lenient with non-mahram men. If the wife's request for divorce is due to harm from the husband, then her request is legitimate, and she is entitled to the financial rights of a divorced woman. However, if it is for reasons other than harm, and she dislikes him, then she may seek a khul' (divorce initiated by the wife) in exchange for a financial compensation, such as forfeiting her mahr (dowry). Minor children have the right to financial support from their father, and their custody belongs to their mother, provided there is no impediment for her. Her right to custody is forfeited if she remarries after her 'iddah (waiting period).