Is it permissible to combine and shorten prayers at home before traveling?
It is impermissible to combine and shorten prayers at home before traveling because a person is called a traveler once he/she departs the borders of his/her country, and offering prayer that way was originally intended to exempt a traveler. And Allah Knows Best.
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.
Is it valid to perform the obligatory prayer while sitting?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Standing (Al-Qiyam) while having the ability to do so is one of the pillars (Arkan) of the prayer; the obligatory (Faridah) prayer is not valid without it. However, whoever is unable to stand may pray sitting down. As for voluntary (Nafilah) prayers, it is permissible to pray them sitting even if one is able to stand, but the one sitting receives half the reward of the one who prays standing. And Allah the Exalted knows best.
What is the ruling of Islamic Law on obtaining financing from Islamic banks by bringing invoices in prior arrangement with the seller, then taking the cash after the financing is approved — with the purpose of settling debts and buying a car?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It appears from the transaction described that its underlying purpose is to obtain cash through resorting to a legal artifice (ḥīlah/tawarruq-style trick), and it is not permissible to engage in such trickery or to coordinate with the various parties merely to obtain liquidity. This is because such artifices bring the sale closer in reality to a ribā-based (usurious) transaction, even if it outwardly appears otherwise.
The Prophet ﷺ said: "Do not commit what the Jews committed, thereby making lawful what Allah has made unlawful through the flimsiest of tricks." This was related by Ibn Baṭṭah in his book Ibṭāl al-Ḥiyal (p. 47), and many scholars have graded it as ḥasan (sound/acceptable).
We have already explained the impermissibility of engaging in such deceptive dealings with Islamic banks in Fatwa No. 516, so please refer to it. And Allah, the Exalted, knows best.