Should the wife's sister be informed that she is an illegitimate child, and should her husband be informed of that, given that there is a birth certificate proving she is the wife's sister, and is she entitled to inheritance?

Adoption is forbidden in Islamic law. If a woman is not a man's daughter, it is not permissible to attribute her lineage to him. No legal rulings arise from this false attribution; thus, she does not inherit from the wife's mother if the latter is not her biological mother. If the adult heirs agree to give her something from their share, it is an act of charity on their part. It is not obligatory to inform her or her husband that she was an abandoned child, but it is necessary to change the lineage and revert the matter to its truth. She can be attributed by a general lineage, such as "so-and-so, daughter of Abdullah."