What is the ruling on a wife who left her husband's home and refused to return, then requested a divorce after being given a choice, and what is the ruling on the maintenance of the child she is raising, and what are the rights of the wife and the child?

It is not permissible for your wife to leave your home without your permission. This is considered an act of disobedience and insubordination (nushooz) that nullifies her right to financial support (nafaqah), unless she is pregnant, in which case there is a difference of opinion among scholars regarding the obligation of her financial support. If you give her the choice between returning or divorce, and she chooses divorce, there is no sin upon her, based on the statement of Aisha (may Allah be pleased with her) that the Prophet (peace be upon him) gave his wives a choice, and they chose him, and this was not considered a divorce. You can refuse to divorce her until she seeks khul' (a form of divorce initiated by the wife in exchange for compensation). The rights of a divorced woman include financial support and housing during the waiting period (iddah) if the divorce is revocable, a compensatory gift (mut'ah), and deferred dowry (mu'akhkhar sadaq) if applicable. As for the child's rights, they include financial support and providing housing until he reaches an age where he can earn a living, as well as his education and care.