Is it permissible to give Zakat al-Mal (obligatory charity on wealth) to a woman who has a small salary that does not cover her living expenses, and who spends on a large family who have no provider other than her?

It is permissible to give Zakat to a woman if her maintenance is not obligatory upon you, and you must ensure that her salary is insufficient to cover basic needs, not luxuries. Remember that the poor who have no source of income are more deserving of Zakat than those who have an insufficient income.