When is a child instructed to fast?
A child is instructed to fast at the age of seven if they are capable of fasting and have reached the age of discernment, by analogy to prayer. They should be encouraged but not forced, so they can become accustomed to it. It is obligatory for their guardian, whether a father or another, to instruct them.
What is the ruling if a postpartum woman becomes pure before forty days; are acts of worship obligatory upon her, and is she permissible for her husband?
If the postpartum woman becomes definitely pure before forty days, she must perform the ritual bath and perform acts of worship as a pure woman does. What was prohibited for her also becomes permissible, so she becomes permissible for her husband after her bath. The minimum duration for postpartum bleeding is a moment (an instant), and its usual maximum is forty days. Reaching forty days is not a condition; rather, it is sufficient for the blood to stop or to see the white discharge (qassa bayda'). And Allah the Almighty knows best.
A person who has recently embraced Islam is afraid of undergoing circumcision (khitān) — what should he do?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
If this person would be harmed by undergoing circumcision, based on the assessment of a trustworthy physician, then he is not obligated to undergo it. However, if the matter is simply a fear of the procedure itself, then it may be postponed until he feels at ease and settled about it, provided this does not lead him to become averse to Islam — while still taking care to remove any impurity from the concealed area [beneath the foreskin]. And Allah, the Exalted, 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.