Praise be to Allah.
What this man did of encouraging the woman to run away from her husband and get divorced from him by means of khul‘ is a haraam action and a grievous sin. It is not permissible for a Muslim to propose marriage to any married woman, or to turn her against her husband, even if that is not with the aim of marrying her and he has no interest in her, no matter what the reason. A stern warning concerning that was narrated in the hadeeth of the Prophet (blessings and peace of Allah be upon him): “He is not one of us who turn a woman against her husband.” Narrated by Abu Dawood, 2125; classed as saheeh by al-Albaani in Saheeh Abi Dawood.
What the woman did of getting divorced from her husband by means of khul‘ with no good reason is also the subject of a stern warning, in the hadeeth in which the Prophet (blessings and peace of Allah be upon him) said: “Any woman who asks her husband for a divorce with no good reason, the fragrance of Paradise will be forbidden to her.”
Narrated by at-Tirmidhi, 1187; classed as saheeh by al-Albaani in Saheeh Abi Dawood.
Because of the crime that they have committed and the abhorrent nature of what they have done, some of the scholars are of the view that if a man turns a woman against her husband, it is not permissible for him to marry her; rather it is permanently forbidden for him to marry her. This is the view of the Maalikis. However, the majority of scholars think that the marriage is valid, despite the sinful nature of their actions.
It is not permissible for this man, or anyone else, to claim the child of another man as his own; rather this comes under the heading of lies and false declarations, transgression against the rights of others and corruption of lineage, and it leads to serious evil consequences. For that reason, Allah has forbidden adoption (in the sense that the adopted child is claimed as one’s own flesh and blood) and claiming the child of another as one’s own, even if the birth parents agree to that or his father is not known. So how about if he usurped the child and falsely claimed him as his own? This is even worse and even more abhorrent.
Allah, may He be exalted, says (interpretation of the meaning):
“Allah has not put for any man two hearts inside his body. Neither has He made your wives whom you declare to be like your mothers backs, your real mothers. (Az-Zihar is the saying of a husband to his wife, ‘You are to me like the back of my mother’ i.e., You are unlawful for me to approach.), nor has He made your adopted sons your real sons. That is but your saying with your mouths. But Allah says the truth, and He guides to the (Right) Way.
Call them (adopted sons) by (the names of) their fathers, that is more just with Allah. But if you know not their fathers (names, call them) your brothers in faith and Mawaleekum (your freed slaves). And there is no sin on you if you make a mistake therein, except in regard to what your hearts deliberately intend. And Allah is Ever Oft-Forgiving, Most Merciful.”
What they must do is fear Allah and repent to Him from these sins: his sin of turning a woman against her husband and her sin of obeying him in that and getting divorced from him by means of khul‘, then his sin of claiming the child of another man as his own, and her helping that evildoer in his evil act and transgression. They should both beware of the wrath and punishment of Allah, and they should realise that Allah is severe in punishment and that He may or may not give a respite. Allah, may He be exalted, says (interpretation of the meaning):
“Consider not that Allah is unaware of that which the Zalimoon (polytheists, wrong-doers, etc.) do, but He gives them respite up to a Day when the eyes will stare in horror”
Muslim narrated in his Saheeh (2583) that Abu Moosa (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Allah, may He be glorified and exalted, grants respite to the wrongdoer, but when He seizes him He will not let him off.” Then he recited: “Such is the Seizure of your Lord when He seizes the (population of) towns while they are doing wrong. Verily, His Seizure is painful (and) severe” [Hood 11:102].
For more information, please see questions no. 84849 and 185184
And Allah knows best.