Is it obligatory to have the intention for each day of fasting, or is one intention sufficient for the whole month?
The intention is obligatory for each day of Ramadan because each day is an independent act of worship separate from the others.
The intention must be made at night before the break of dawn, as the Prophetﷺ said: "Whoever does not intend fasting at night, there is no fast for him." [An-Nasa’i]
And he also said: "Whoever does not firmly resolve to fast before dawn, there is no fast for him." [At-Tirmidhi, Abu Dawood, and An-Nasa’i]
Whoever wakes up and eats Suhoor while mindful of fasting has made the intention. Likewise, one who firmly intends at any moment during the night to fast the next day has also fulfilled the intention.
Is it permissible for the wife to give the Zakah (obligatory charity) of her money to her poor husband?
Yes, it is permissible for the wife to give from the Zakah of her money to her poor husband. And Allah Knows Best.
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.
What is required of one who doubts the number of rakʿāt during prayer?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
If a person doubts the number of rakʿāt he has prayed, he should build upon the lesser number, as that is what he is certain of. He should then perform the prostration of forgetfulness (sujūd al-sahw) before the final salām at the end of the prayer. ʿAṭāʾ ibn Yasār narrated that the Messenger of Allah ﷺ said: "When any one of you is in doubt during his prayer and does not know whether he has prayed three rakʿāt or four, let him pray one more rakʿah and then perform two prostrations while seated before the salām. If the rakʿah he prayed was a fifth, these two prostrations will make it even; and if it was the fourth, then the two prostrations serve as a humiliation for the devil." (Reported by Abū Dāwūd.)
It is stated in al-Muqaddimah al-Ḥaḍramiyyah — one of the foundational texts of the Shāfiʿī school: "If one doubts whether he has prayed three rakʿāt or four, he is obliged to build upon the lesser number."
However, if such doubt recurs repeatedly and reaches the level of obsessive whispering (waswasah), he should not build upon the lesser number in that case — rather, he should build upon the greater number. And Allah the Almighty knows best.