Is the aforementioned marriage permissible in Islamic law, what are the requirements for a valid marriage, and what is the ruling on my two children born from this marriage?

Marriage in Islam has pillars and conditions; among them are the presence of a guardian (wali) and two just Muslim witnesses. It is permissible for a Muslim man to marry a Kitabiyyah (a woman from the People of the Book). If the husband is a Christian, her guardian is her closest male relative (asabah) who shares her religion. If such a guardian is not available or refuses, the Muslim judge can marry her off. However, if the husband is a ترک صلاة (one who abandons prayer), his marriage to a Christian woman is considered a contract of an apostate. But if both spouses embrace Islam, their marriage is affirmed, and they do not need to renew the contract. The child resulting from such a marriage is legally attributed to his father, even if the original marriage was invalid, because "the child belongs to the bed."