What is the ruling on a person in a state of major impurity (junub) or a menstruating woman (ha'id) reciting the Quran from memory?
It is not permissible for a menstruating woman, a postpartum woman, or a person in a state of major impurity to recite anything from the Quran, whether from memory, from the Quran, from a phone, or a computer. It is also not permissible for them to touch the Quran, based on what was reported from Ali bin Abi Talib that the Prophet (peace be upon him) was not prevented from anything regarding the Quran except major impurity (janabah). (Reported by al-Tirmidhi who said it is a hasan sahih hadith). Menstruation and postpartum bleeding are analogous to major impurity (janabah) as they are all major impurities (hadath akbar).
For those mentioned, it is permissible to mention Allah and supplicate even with verses from the Quran, provided they do not intend them as recitation of the Quran, but intend them as remembrance (dhikr) or supplication (du'a). And Allah the Almighty knows best.
What is the ruling on giving the expiation of an oath (Kaffarat al-Yamin) to a charity that feeds the poor?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is allowed for a Muslim to appoint a proxy (agent) to distribute the expiation (Kaffarah) on their behalf. Our jurists have explicitly stated the permissibility of delegation (Tawkil) in the distribution of Zakat, Kaffarah, and vows (Nadr).
It is stated in Mughni al-Muhtaj (Vol.3/P.237): 'One has the right to pay the Zakat of their wealth personally... and they also have the right to delegate it.' And Allah the Exalted knows best.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
What is the ruling on swearing on the Holy Quran and breaking that oath?
It is impermissible to swear on the Holy Quran in vain, and it is also impermissible to subject it to such an act since breaking the oath after swearing on the Quran entails an expiation.