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.
How is Laylat al-Qadr observed?
Laylat al-Qadr is observed through various acts of worship and obedience, including:
● Performing obligatory, Sunnah, and voluntary prayers
● Reciting the Quran
● Increasing remembrance (dhikr) of Allah
● Giving charity
● Performing I‘tikaf
● Inviting others to Allah
It is also Sunnah to increase in supplication and humbly beseeching Him, Glorified and Exalted be He.
Does buying meat equivalent to the weight of a slaughtered animal and distributing it avail for an udhiyah?
Praise be to Allah, and prayers and peace be upon our Master the Messenger of Allah.
That does not avail for the sacrifice (uḍḥiyyah), because the sacrifice necessarily requires slaughtering an animal from the category of anʿām (camels, cattle, and sheep/goats) within a specified time.
Rather, that is considered a form of charity (ṣadaqah) for which one is rewarded. And Allah Almighty knows best.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.