How does a praying person prostrate?
All perfect praise be to Allah,The Lord of The Worlds It is from Sunnah that he/she sits with both knees on the ground in order to prostrate, then hands, nose and forehead are placed on the same place. In addition, the toes have to touch the ground, and be directed towards the Qiblah (direction of Ka`bah) during prostration. And Allah Knows Best.
Is it permissible for the mother to offer an Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)for her son?
Aqeeqah is due on the one obliged to provide for the newborn, and it is permissible for the grandfather, or the mother to offer the Aqeeqah.
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
Is it permissible for a person to sever ties with his sister because her husband had quit prayer and done him harm?
A sister is a blood-relative and maintaining blood-relations is obligatory in Islam. As regards her husband, he should be invited to the way of Allah with wisdom and beautiful preaching.