Is it permissible to slaughter a single sheep with the combined intention of both the uḍḥiyyah and the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
According to Shaykh al-Islām Imam Ibn Ḥajar al-Haytamī, it is not permissible to combine the intention of the uḍḥiyyah and the 'aqīqah in a single animal, as each of the two has a distinct and separate cause that differs from the other.
However, Shaykh al-Islām Imam al-Ramlī permitted the combining of both intentions in a single animal — and this position offers a degree of latitude and ease. And Allah Almighty knows best.
Is it permissible for a mother to stop her daughter from marrying a respectable, God-fearing young man because he isn't college/university graduate?
It is impermissible to deny a woman her right in getting married except for a lawful reason, and problems can be solved through dialogue. However, she can file a suit in order for the judge to have a say in that regard.
Is it valid to perform Tarawih prayer at home?
It is Sunnah for a Muslim to perform Tarawih prayer in congregation at the mosque.
However, if a person sometimes prays it at home in congregation with their family for a valid reason, there is no harm in doing so.
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.