Is it permissible to marry a Christian woman who verbally converted to Islam for the purpose of marriage, without a genuine intention to embrace Islam? And is it valid for the judge to officiate the marriage contract in this case?

It is not permissible for you to cohabit with her. The judge's ruling does not make lawful what is unlawful. You must repent to Allah for the lie. This marriage, without a guardian, is annulled according to the majority of scholars. You may contract a new marriage with her with the consent of her guardian. The children from this marriage are your children. The ease of religion does not mean leniency in the Sharia rulings.