Does nosebleed during the day in Ramadan affect the validity of fasting?
A nosebleed does not affect the validity of fasting unless some of the blood reaches the body cavity or the person intentionally swallows it. In such a case, their fast is invalid, and they must refrain from eating and drinking for the rest of the day and make up for that day later.
Is Fajr (dawn) Sunnah a confirmed one?
All praise be to Allah The Lord of The Worlds Of course, it is a confirmed Sunna which the Prophet (PBUH) used to offer in a regular basis. And Allah Knows Best.
Is bleeding that lasts for nine or ten days considered menstruation (Hayd)?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
If the duration of the bleeding does not exceed fifteen days, then all of it is considered menstruation (Hayd), even if the regular cycle ('Adah) has changed. 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.