Is it permissible to have an urfi marriage with a non-Muslim woman and for her to become pregnant, given her fear of her family and her desire to embrace Islam, especially since the questioner's relationship with his family is severed, he suffers from illnesses, and he desires to have children?

A Muslim must learn what he is ignorant of concerning his religion, especially regarding marriage, for Allah the Almighty says: "So ask the people of the message if you do not know."

If the disbelieving woman is not from the People of the Book, then your marriage to her is invalid by consensus, and it is not permissible to marry polytheistic women until they believe. If she embraces Islam, then marriage to her becomes permissible.

And if what you called an " عرفي (customary) marriage" is merely an agreement between you without a guardian (wali) or witnesses, then it is fornication and not a marriage. However, if the marriage is to a woman from the People of the Book, it is permissible provided she is chaste, and the Muslim judge marries her off if she does not have a Muslim guardian. If a disbelieving woman embraces Islam and does not have a Muslim guardian, the Muslim judge marries her off; if no judge is available, she entrusts a trustworthy Muslim man to marry her off.