What is the ruling of Islamic law regarding a man marrying a woman without a legitimate (Sharia-compliant) contract, while she is aware of his first wife's disapproval, and what is the ruling on their living in the same house?

If the marriage was conducted with a valid Sharia contract, such as the presence of the guardian, two just witnesses, and a formula indicating the contract, then the marriage is valid. It does not require reciting Al-Fatiha or the approval of the first wife. Reciting Al-Fatiha in a marriage contract has no basis in Islamic law.

However, if the matter was limited to merely reciting Al-Fatiha and a wedding celebration, then the marriage was never validly contracted in the first place. The woman remains a stranger (non-mahram), and it is forbidden to seclude oneself with her or touch her. Their living together is a grave transgression (munkar). If cohabitation occurs between them, it is considered adultery (zina), and they must repent. After repentance and ensuring the woman is free of pregnancy (istibra'), they can marry with a contract that fulfills all conditions.