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.
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 to shorten the eyebrows if they get too long?
Yes, it is permissible, but not desirable.
Is it permissible for a woman who broke fasting due to delivery to pay a ransom ?
She should make up for the missed fasting days once she becomes ritually pure, but it is impermissible for her to pay a ransom while being able to fast.