What is the ruling on a man marrying a thrice-divorced woman with the intention of making her lawful for her first husband, without his knowledge? And does she thereby become lawful for her first husband? And is this considered part of the marriage of a "muhallil" and the one for whom it is made lawful?

It is not permissible for a man to marry a woman with the intention of making her lawful for her previous husband, even if the husband is unaware of this. This falls under the category of the forbidden marriage of tahlil (making lawful), which is the view of the majority of scholars. Ibn Abi Shaybah mentioned in his Musannaf several statements from the Successors rejecting this type of marriage.

Shaykh al-Islam Ibn Taymiyyah clarified that the marriage of al-muhallil (the one who makes lawful) is invalid and forbidden and does not make the woman lawful for her first husband. If a man marries her with the intention of divorcing her so that she can return to her previous husband who divorced her three times, then this marriage is unlawful and invalid, whether this was agreed upon verbally or implicitly, and whether the first husband knew or not. This includes the case where the woman and her guardian are unaware of the intention of tahlil.

The correct view is that the woman does not become lawful for her first husband who divorced her three times unless she marries a man in a genuine marriage of desire, and he consummates the marriage with her, and then they separate by death, divorce, or annulment. Only after that may her first husband marry her.