When is Istikharah (guidance prayer) Du`a offered, before Tasleem (saying asalamu alikum warahmatu allah wa barakatuh to end the prayer), or after it?
It is offered after Tasleem.
Does smoking break the fast?
Yes, smoking breaks the fast because smoke particles intentionally enter the lungs, which are considered part of the body cavity (jauf).
Who is responsible for performing the aqiqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The one upon whom the 'aqīqah is incumbent is whoever is obligated to financially maintain the newborn — whether the father, paternal grandfather, or mother. It is to be performed from their own wealth, not from the wealth of the newborn child.
As for one upon whom the financial maintenance of the newborn is not obligatory, they may not perform the 'aqīqah except with the permission of the one upon whom it is obligatory — namely, the father. And Allah Almighty knows best.
What is the ruling on mentioning Allah`s name upon slaughtering an animal?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is Sunnah for the slaughterer to say at the time of sacrificing the Udhiyah: (Bismillah - In the name of Allah); due to the saying of the Almighty: "So eat of that [meat] upon which the name of Allah has been mentioned" [Al-An'am/118]. It is not obligatory; if one leaves it out intentionally or forgetfully, the consumption of the meat remains permissible. And Allah the Almighty knows best.