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 woman to shake hands with her brother-in-law?
It is forbidden for the woman to shake hands with her brother-in-law because he isn`t a Mahram(unmarriageable) of hers.
Is it permissible to pay the expiation of an oath to a charitable party as it feeds the poor and needy such as Tkiyet Um Ali?
All perfect praise be to Allah, The Lord of The Worlds, and May His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.
There is no harm on authorizing someone to handle the expiation of another to the poor and needy, since Shafiee's jurists stated that authorizing someone for the purpose of distributing Zakah, expiation and vow is permissible as stated in [Mughni Al-Muhtajj, vol.3 pp.237]. At last, the intention of the person who intends to authorize someone else suffices in this regard. And Allah Knows Best.
Is it permissible for a Muslim woman to wear trousers at home?
Is it permissible for the man to put on revealing clothes? Answer: It is impermissible for the man to put on revealing clothes that show his Awrah(private part) which is between his navel and knee, and exercising modesty is better.