Is the marriage of a divorced woman to a married man secretly, without the knowledge of his first wife and her family, valid, and does it waive her right to inheritance in exchange for financial compensation?

There is no deficiency or error in your marriage if it fulfilled the Shariah conditions. Your husband is not entitled to prevent you from inheritance after his death; rather, your right to the estate remains valid, and you can claim it. As for your husband favoring you by buying a house, it is permissible according to the majority of scholars. Your husband should be gentle with his mother and strive to inform her of the marriage in a way that does not anger her, explaining that this is from Allah's law.