Is the niṣāb calculated based on the price of 21-karat gold, or 24-karat gold?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Zakāh is obligatory on cash and on gold designated for saving or for trade, provided the total reaches the niṣāb (minimum threshold liable for zakah), and a full ḥawl (lunar year) has elapsed over it. The niṣāb is (85) grams of 24-karat gold; for 21-karat gold, the niṣāb equals (97) grams; and for 18-karat gold, it equals (113) grams.
The relied-upon position is to calculate the zakāh based on the buying price [i.e., the price the gold merchant would pay the owner to purchase the gold from him] on the day the zakāh obligation is due.
The one paying zakāh may assess/calculate the niṣāb using any of the aforementioned gold karats [i.e., he may choose whichever karat standard he wishes for the purpose of the calculation]. And Allah, the Exalted, knows best.
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, but not an udhiyah. And Allah Almighty knows best.
What is the meaning of the word aqiqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The 'aqīqah in its linguistic sense refers to the hair upon the head of a newborn at the time of birth.
In its Sharī'ah sense, it refers to the livestock that is slaughtered on behalf of a newborn child.
And Allah Almighty knows best.
Is it permissible for a woman to observe voluntary fast without her husband's permission?
A woman is not permitted to observe a voluntary fast (nafl) while her husband is present without his permission.