I hired a private tutor for my son through an academy that sends teachers [to families] and takes a percentage [of their pay]. At the start of the [new] school year, I arranged directly with that same teacher, without the academy's knowledge. What is the ruling on this?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The ruling on this matter depends on knowing the specific terms of the agreement and contract concluded with the academy. If the agreement stipulates that no arrangement may be made with any of the students except through the academy itself, then it is not permissible for you to breach the terms of this contract, based on the words of our master, the Messenger of Allah ﷺ: "Muslims are bound by their conditions." (Related by al-Bukhārī)
However, if the agreement pertains only to teaching a specific subject, and the agreement concludes once that subject is finished, then there is no objection to arranging with the teacher to teach other subjects after the contract period has ended, outside of the academy.
Whatever the case maybe, the teacher must uphold the terms of his contract with the academy. And Allah, the Exalted, knows best.
Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. And Allah Knows Best.
What is the ruling on giving Zakat al-Fitr in cash?
The default ruling is that Zakat al-Fitr should be given as the staple food of the land. In Jordan, for example, the staple food is wheat or rice, and the amount of Zakat al-Fitr is 2,500 grams per person. It is easy to give this amount of rice to the poor and needy, and this is the correct ruling according to all Islamic schools of thought.
However, Hanafi scholars have permitted giving Zakat al-Fitr as monetary value, considering it more beneficial for the poor and easier for the giver.
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.