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.
What is the ruling on the Friday bath (ghusl al-Jumu'ah)?
The Friday bath is a confirmed Sunnah, based on the Prophet's (peace be upon him) saying: "Whoever performs ablution on Friday, it is good for him, and whoever takes a bath, bathing is better." (Reported by Abu Dawud and al-Tirmidhi who said it is a hasan hadith). It is recommended for everyone who wants to attend the Friday prayer, even if they are not from its usual attendees, like a woman or a boy.
If a person is in a state of major impurity (junub) and intends by his bath both lifting the major impurity and the Friday bath, he achieves both. If he intends only one of them, he achieves only what he intended. And Allah the Almighty knows best.
What are the conditions of the Aqeeqah?
The animals that are acceptable to be slaughtered for 'Aqeeqah are the same which are acceptable to be slaughtered for the sacrifice of 'Eed in terms of their age and lack of defects. The sheep should be one year old, and its teeth should have fallen off. As for goats and cows, they should have finished two years of age and entered into the third. The Aqeeqah should be free of any defects as regards meat, or price value.
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.