Is it permissible for one to shake hands with his uncle`s daughter?
No, it isn`t because he is a non-Mahram (Marriageable) to her.
What is the ruling on the white discharge (*Al-Qassat al-Bayda'*) if it is in a very small amount—is it considered a sign of purity?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Among the signs of purity (Tahr) for a woman is the white discharge (Al-Qassat al-Bayda'), which is a clear, pure white fluid, regardless of whether it is minimal in quantity, or complete dryness (Jafaf), wherein secretions cease entirely such that if a white cotton cloth is inserted, it comes out completely clean. Once a woman observes either of these two signs, she has attained ritual purity.
Furthermore, another sign of purity is when the duration of the menstrual period exceeds the maximum limit of menstruation, which is fifteen days. If it exceeds this period, it becomes evident that she is experiencing non-menstrual bleeding (Istihadah).
What is the ruling on offering an Udhiyah on behalf of another with their permission?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible for a person to offer a sacrifice (Udhiyah) on behalf of someone else with their permission, even if the person performing it has not offered a sacrifice for themselves. It is stated in Sharh Manhaj al-Tullab (Vol.5/P.261) by Zakariya al-Ansari: "No one may offer a sacrifice on behalf of another without their permission... as opposed to when permission has been granted." And Allah the Almighty knows best.
What is the ruling of Islamic Law on shaving body hair and clipping the nails for one who is in a state of major ritual impurity (*junub*)?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is recommended for one in a state of major ritual impurity (junub) not to remove anything from his body until after the ritual bath (ghusl). It is stated in Bushrā al-Karīm bi-Sharḥ Masāʾil al-Taʿlīm (p. 135): “It is recommended for one in a state of janābah and the like not to remove anything from his body until after ghusl, because the parts of the body return to him in the Hereafter, and he returns in a state of janābah, as a rebuke to him. Then it is removed from him, except for the original parts. It is said that every hair demands its due for its janābah.”
It is stated in Ḥāshiyat al-Shabrāmallisī ʿalā Nihāyat al-Muḥtāj (Vol.2/p.341): “If he wishes to combine shaving and the Friday ghusl, he should delay the shaving until after the ghusl if he is in a state of janābah, so that the ghusl removes its trace from the hair.” And Allah Almighty knows best.