What is the ruling on someone who eats or drinks thinking that the sun has set, then realizes that it has not yet set?
Whoever eats or drinks believing that the sun has set, then later discovers that it has not yet set, their fast is invalid, and they must make up that day after Ramadan. It is not permissible to break the fast before confirming sunset—either by seeing it, through personal reasoning, or by relying on the statement of someone trustworthy in their religious commitment.
What is the ruling on wiping the front of the head beneath the ḥijāb, and is it permissible to wipe over the ḥijāb if it was put on while in a state of purification (wudu`)?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
It is permissible to wipe the front portion of the head (nāṣiyah) with the fingertips. It is likewise sufficient to place a wet hand over a head covering (ḥijāb), provided the moisture actually reaches the hair beneath it — if it does not, it does not suffice. A head covering is not treated in the same manner as leather socks (khuff) and may not be wiped over in lieu of the head itself.
Imām al-Nawawī, may Allah have mercy upon him, states in al-Majmūʿ (Vol.1/P.407): "If a person is wearing a turban and does not wish to remove it — whether for a valid reason or otherwise — he should wipe the entire front portion of the head... The same ruling applies to whatever a woman wears on her head. If, however, he confines himself to wiping over the turban without wiping any part of the head itself, this does not suffice — and there is no disagreement among us on this point." He further states: "A woman is like a man in the manner of wiping the head... She should insert her hand beneath her head covering so that the wiping falls upon the hair itself. If she places her wet hand over her head covering, our scholars stated: if the moisture does not reach the hair, it does not suffice her." And Allah the Almighty knows best.
What is the ruling on sacrificing a castrated animal or one with a missing tail?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible to sacrifice a castrated sheep; as it has been established that the Prophet (peace be upon him) sacrificed:
"...two large, fat, horned, white-and-black, castrated rams (Mawju'ayn—meaning having crushed testicles)." [Narrated by Ibn Majah in his Sunan].
It is not permissible to sacrifice an animal that is missing its tail, or udder due to being cut off. This is in contrast to an animal that was naturally born without a tail, or udder; such an animal is valid for sacrifice. And Allah the Almighty knows best.
Is it permissible for the mother to give her children from the Zakah (obligatory charity) of her money?
It is permissible for the mother to give her children from the Zakah of her money if they are Zakah-eligible recipients, and she isn`t obliged to provide for them.