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?
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 hair extension permissible?
All Perfect Praise is due to Allah, The Lord of The Worlds
It is impermissible for the man and the woman to apply a human`s hair as extension, but using artificial hair is permissible for the married woman after obtaining the approval of her husband so long as non-Mahrams don`t see her. And Allah Knows Best.
Is it permissible for a woman to sit with her brothers-in-law?
It is impermissible for a woman to sit with her brothers-in-law, and it is also impermissible for a person to have a Khulwa (seclusion) with his sister-in-law.
Is it permissible for a woman to swear a false oath by the Holy Quran to please her distrustful husband?
It is forbidden for a person to do so since this is considered a false oath (Yamaan Ghamoos) which dips (Yaghmos) its taker in Hellfire if he/she doesn`t seek repentance, and he/she has to expiate for that oath. You should avoid any doubtful acts so that your husband`s chest remains clear from the evil suggestions of the devil.