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 for a Muslim to slaughter an Aqeeqah on behalf of someone else, and offer it to him as a gift?

In principle, the guardian is the one who should offer the Aqeeqah(the sheep slaughtered on the seventh day from the child`s birth) because he is obliged to provide for the newborn, and it is impermissible for anyone else to slaughter it on his behalf unless with his consent. However, it is permissible for a person to offer the sheep, or its price as a gift to the guardian of the newborn, and then the latter can slaughter it, or deputies someone else to do that on his behalf.

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.

For passing time, my friends and I have called a famous diviner. She told me that somebody had cast a spill on me not to get married for the rest of my life. I didn`t believe her, but I can`t forget her words. How can I protect myself from such thing?

Praise be to Allah the Lord of the Worlds.

Dealing with diviners is forbidden and what you are suffering from is due to your having consulted that diviner. Have trust in Allah and ask Him for forgiveness, but rest assured that whatever Allah has decreed will take place. And Allah The Almighty Knows Best.