Is a wife considered sinful if she asked for divorce because her husband married another woman?
A second marriage is lawful, and so it is impermissible for you to ask for divorce because of that. If you are certain that you won`t be able to deliver Allah`s rights and those of your husband`s , then it is permissible for you to ask for Khul`a (Divorce on the demand of the wife). However, you should exercise patience since after difficulties, there is easiness by the will of Allah.
Is it permissible for a wife to leave her house without asking her husband?
It is impermissible for the wife to leave her house without asking her husband except for a sound reason.
What is the ruling on giving Zakah (obligatory charity)to one`s relatives?
It is impermissible to pay Zakah to one`s origins (parents and grandparents) because providing for them is an obligation on their branches (sons and daughters) if they were poor, but it is permissible to give some of the Zakah money to relatives whose provision isn`t due on the giver of the Zakah.
I have mixed (halal and haram) money, and I want to purify it. My father is poor, and I want him, my mother, and my mother-in-law to perform Hajj, along with myself and my wife, as they need care due to their old age. Is it permissible to use this money for Hajj, or what should I do with it? Please advise.
If you knew the exact amount of unlawful money, then you should give as a charity, and if didn't then try to figure out the closest amount in order to give it as a charity.
As for performing Hajj, perform it by using your lawful money. May Allah Bless you for being dutiful to your parents and your mother in law and for your attempt to serve them, And Allah Knows Best.