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.
What is the ruling on swearing on the Holy Quran and breaking that oath?
It is impermissible to swear on the Holy Quran in vain, and it is also impermissible to subject it to such an act since breaking the oath after swearing on the Quran entails an expiation.
Does the intention (niyyah) of the principal (muwakkil) suffice for the sacrifice (dhabḥ) of the udhiyyah?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
If a person appoints an agent (wakīl) to perform the slaughter (dhabḥ) on their behalf, the intention (niyyah) of the principal (the one appointing the agent) suffices, and there is no need for the agent to have this intention himself. In fact, even if the agent is unaware that he is performing the sacrifice on someone's behalf as an udhiyyah (sacrificial offering), this does not affect its validity. And Allah, the Most High, knows best.
Should missed Sunnah prayer be made up?
Making up missed voluntary acts of worship is from Sunnah.