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 it permissible to slaughter a sheep on Eid al-Adha with the intention of charity (sadaqah) on behalf of a sick person, and with the intention of udhiyah (sacrificial animal) at the same time?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The udhiyah (sacrificial animal) is an act of worship intended for its own sake; therefore, it must be slaughtered with the intention (niyyah) of udhiyah.
Whoever intends to draw closer to Allah through this act of obedience with the intention of udhiyah, and also intends to gift the reward to another person, there is no Islamic objection to that. And Allah Almighty knows best.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.
What is the ruling on making up missed fasts from many years?
A Muslim must make up for any missed fasts, as they are a debt owed to Allah. The Prophet ﷺ said:
"The debt owed to Allah is more deserving of being fulfilled." [Narrated by Muslim]
Whoever has missed fasts must make them up as long as they are alive and capable of fasting.