Referring the question to Dr. Main K. Al-Qudah a Member of the Assembly of Muslim Jurists in America,
1- The apostasy status of the ex-husband can not be declared only because of his denial of the Sunnah. He could be an ignorant person who needs education, or was persuaded by a misguided person while thinking he is a scholar, or for any other reason. In other words; only a Mufti or a high religious authority is allowed to declare him as an apostate.
2- After Khlu`, he is still responsible for the sponsorship of his kids as long as they need his support. A lawyer should be consulted as well regarding the children because of the legal procedures.
3- If anyone asked the woman after Khlu’ about her ex-husband, she should find out the reason behind the question; is it for marriage purposes, or business partnership, ..etc, then she can answer by giving a general statement that is sufficient for the inquirer to understand that she does not recommend this individual.