Is it permissible to purchase the Udhiyah on installments?
It is permissible to purchase the sacrificial animal (uḍḥiyyah) on installments or by borrowing its price. However, it is not recommended for the poor to do so, because they are not required to offer a sacrifice, and Allah does not burden a soul beyond its capacity. If doing so would lead to negligence in providing for one's dependents (nafaqah wājibah), then giving precedence to obligatory maintenance over borrowing to buy the sacrifice is appropriate. This is based on the saying of the Prophet (peace and blessings be upon him): "It is sufficient sin for a person to withhold food from those whom he is responsible to support."
Furthermore, the sacrifice is valid from one who has a debt, but it is preferable (awlā) to repay the debt first, especially if the debt is due immediately (ḥāll). And Allah Almighty knows best.
Does an internal medical examination for a woman affect her fast?
An internal medical examination for a woman invalidates the fast because it involves the entry of a foreign object into the body cavity (jauf) while fasting.
In this case, the woman must refrain from eating and drinking for the rest of the day out of respect for Ramadan and make up for the missed fast after Ramadan.
Such an examination should be avoided in Ramadan and other months unless absolutely necessary, as exposing the private parts is only permissible in cases of necessity.
If necessary, a woman should seek a Muslim female doctor first. If one is unavailable, she may see a female doctor from the People of the Book (Jews and Christians). If neither is available, she may consult a trustworthy and competent Muslim male doctor.
What is the ruling on the fasting of a woman who has reached menopause if menstrual blood flows?
If a woman reaches the age of menopause (which is usually sixty-two) and her menses have ceased, then she sees blood after that, and its duration is not less than a day and a night (24 hours), it is menstruation (hayd). If it is less than a day and a night, she is considered as having non-menstrual vaginal bleeding (mustahada), so she fasts and prays. However, she must perform ablution for every obligatory prayer after its time enters, pray immediately, and be treated as a person with a continuous condition. There is no specific end limit for a woman's menstruation; it is possible as long as the woman is alive. And Allah the Almighty knows best.
What is the ruling of Islamic Law on a person who abandons prayer after reaching puberty—is he required to make them up?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Making up missed obligatory prayers is an Islamic duty just as performing them in their proper time is obligatory, and taking care of them takes precedence over performing voluntary prayers. This is established by the practice of our Supreme Prophet (peace and blessings be upon him) when the polytheists distracted him on the Day of the Trench from performing the Asr prayer, so he made it up afterward, as well as the Hadith: "Pay Allah what is due to Him, for Allah is most deserving of fulfillment." (Recorded by Al-Bukhari)
Furthermore, it is valid to make up missed prayers at any time.
Whoever has missed a number of prayers and does not know their exact count, it is obligatory upon them to sincerely repent and exert their utmost effort to estimate their number. If they are unable to determine the exact count, they should base it on what is most probable according to their best judgment. They should then make them up gradually; if one performs one missed prayer alongside every current obligatory prayer, the matter becomes easy and manageable, and with the passage of time, one will complete what is owed, Allah willing. And Allah Almighty knows best.