Is it permissible for a woman to uncover her face while performing Umrah (minor Hajj)?
The woman while in a state of Ihram (ritual consecration) is obligated to uncover her face and hands, but at the same time permitted to let her head-covering garment drape from her head down over her face when non-Mahram (i.e., marriageable) men pass by her. And Allah Knows Best.
Is it permissible to slaughter one animal as a sacrifice and an Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)?
It is impermissible to do so since each of them is slaughtered for a different reason.
What is the Islamic ruling on one who was unable to fast and then regained the ability?
He is not required to make up the fast (Qada) even if he becomes capable of it; whether he regained the ability to fast after paying the fidya (feeding a needy person for each day of missed fasting) or before it, because he was liable for paying it in the first place, so it remains binding upon him. However, if he delayed paying it beyond the first year, nothing is required of him due to the delay. If he is unable to pay it, it does not remain as a debt upon him. And Allah the Exalted knows best.
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.