What is the ruling on what the mother did by changing a picture and a name in the passport to perform the Hajj pilgrimage, and is she sinful for that, and is this act considered unlawful?

Forging a passport is forbidden except in cases of dire necessity. It is not permissible to help your father forge it, due to the Almighty’s saying: "And do not cooperate in sin and aggression," and due to the Prophet’s (peace be upon him) saying: "There is no obedience to a human in disobedience to Allah." However, if the father threatens his wife with divorce, and this divorce would be the third, then this is considered coercion, which permits her to do what her husband dislikes.