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 Sunnah to cook the meat of the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah not to distribute the meat of the 'aqīqah raw. Rather, it is recommended to cook it with something sweet — such as raisins or honey — as a good omen and expression of hope for the sweetness of the newborn's character and conduct throughout their life. And Allah Almighty knows best.
What is the ruling on one who doubts washing a limb before or after finishing ablution?
If a person performing ablution doubts leaving the washing of a limb from the limbs of ablution during his ablution, he must repeat washing that limb and wash what comes after it. If he doubts leaving the washing of a limb after finishing the ablution, there is nothing upon him. And Allah the Almighty knows best.
What should the one performing the slaughter say at the time of slaughtering the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah to say the following supplication at the time of slaughtering the 'aqīqah:
"Bismillāhi wallāhu Akbar, Allāhumma laka wa ilayka, hādhihi 'aqīqatu fulān."
("In the name of Allah, and Allah is the Greatest. O Allah, this is for You and unto You — this is the 'aqīqah of [the child's name].")
This is based on what al-Bayhaqī narrated in Al-Sunan al-Kubrā (Vol.9/P.511), that the Prophet ﷺ said: "Slaughter it in his name and say: In the name of Allah, and Allah is the Greatest. O Allah, this is for You and unto You — this is the 'aqīqah of so-and-so."
And Allah Almighty knows best.