What is the ruling on wiping the white area behind the ears instead of wiping the head?
The obligation in ablution is to wipe some of the head; it is not a condition to wipe all of it. Wiping the white area behind the ears suffices for that. And Allah the almighty knows best.
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 the white discharge (*Al-Qassat al-Bayda'*) if it is in a very small amount—is it considered a sign of purity?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Among the signs of purity (Tahr) for a woman is the white discharge (Al-Qassat al-Bayda'), which is a clear, pure white fluid, regardless of whether it is minimal in quantity, or complete dryness (Jafaf), wherein secretions cease entirely such that if a white cotton cloth is inserted, it comes out completely clean. Once a woman observes either of these two signs, she has attained ritual purity.
Furthermore, another sign of purity is when the duration of the menstrual period exceeds the maximum limit of menstruation, which is fifteen days. If it exceeds this period, it becomes evident that she is experiencing non-menstrual bleeding (Istihadah).