Is it valid for the 'aqīqah to be performed using the newborn child's own wealth?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is not permissible for the guardian to perform the 'aqīqah using the newborn child's own wealth, because the 'aqīqah is a voluntary act of giving (tabarru'), and a guardian is prohibited from making voluntary expenditures from the child's wealth. Should he do so, he becomes financially liable for what he spent. And Allah Almighty knows best.
What is the ruling on seeking forgiveness between the two Friday sermons?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to seek forgiveness (Istighfar) between the two Friday sermons (Khutbatayn), and it does not invalidate the Friday prayer (Salat al-Jumu'ah). Furthermore, it is not disliked (Makruh) to speak before the sermon, after it, or between the two sermons. Likewise, it is not disliked for one who enters during the sermon to speak if there is a need for it, provided he has not yet taken a place and settled into it. And Allah the Exalted knows best.
Is it required for a discerning child to make the intention at night if they wish to fast?
If a discerning child wishes to fast during Ramadan, they must make the intention at night because intention is one of the pillars of fasting, even though fasting is not obligatory for them.
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.