What is the ruling of Islamic Law on a menstruating woman (ḥā'iḍ) reciting the Islamic ruqyah (Qur'anic/spiritual healing)?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is permissible for a person in a state of janābah (major ritual impurity), or a menstruating woman (ḥā'iḍ), or a woman in postpartum bleeding (nufasā'), to recite for the purpose of ruqyah (spiritual/Qur'anic healing), even if this includes some verses of the Noble Qur'an — whether recited from memory or by looking at the Muṣ-ḥaf (the physical copy of the Qur'an) without touching or carrying it — provided that the intention behind the recitation is ruqyah, protection (taḥṣīn), and remembrance (dhikr), not the intention of tilāwah (Qur'anic recitation as an act of worship in itself). This remains permissible even if what is recited is an entire sūrah, such as Sūrat al-Mulk.
Imam al-Khaṭīb ash-Shirbīnī (raḥimahullāh) said: "It is lawful for a person in janābah to utter its [the Qur'an's] remembrances and other content — such as its admonitions, its narratives, and its rulings — so long as it is not with the intention of [reciting] Qur'an. [This is] like his saying upon mounting [a vehicle]: 'Glory be to Him Who has subjected this to us, and we could never have accomplished this [by ourselves]' [Sūrat az-Zukhruf: 13] — meaning, [could never have] been capable of it — or upon a calamity: 'Indeed we belong to Allah, and indeed to Him we shall return'... In fact, my teacher, ash-Shihāb ar-Ramlī, issued a fatwā that even if he recited the entire Qur'an without the intention of [reciting] Qur'an, it would be permissible." [Summarized from Mughnī al-Muḥtāj, Vol. 1/P.217] And Allah, the Exalted, knows best.
What should a person, who doubts the validity of his Wudu, or over performs it, do?
One who doubts the validity of his/her Wudu while performing it should redo it until he/she is certain of having attained purity. But, if doubt was after having performed Wudu, then he/she should not pay attention to that as doubt after the end of an act of worship does not count. This is of course in case doubt was within the reasonable limits since once it goes beyond that, it becomes a whispering of the Shaytaan (devil) which he/she should ignore as the Wudu is valid.
Is it permissible to pay the Zakah (obligatory charity) for settling the debt of a deceased relative?
It is impermissible to pay the Zakah for settling the debt of the deceased since Zakah is paid to eligible recipients who are alive. As for the debts of the deceased, they are to be settled from the estate before dividing it amongst the heirs, but if the deceased didn`t leave an estate, then it is desirable for his/her heirs to settle those debts if they can afford them. And Allah Knows Best.
Is it permissible for a woman to ride a taxi without a Mahram (unmarriageable kin) in order to attend a gathering of knowledge?
It is permissible for the woman to pursue the knowledge that she needs in order to perform the religious requirements due on her if there was nobody to teach her at home, provided that there is no temptation involved. Moreover, we don`t recommend that she rides a taxi without a Mahram although such an act isn`t from the forbidden Khulwah (seclusion).