Is it permissible for household members to pool money for the Udhiyah and gift it to one of them?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is permissible for members of a household to pool their money together and gift the total amount to one individual among them so that he may offer the sacrifice (Udhiyah). In this case, the family members receive the reward for giving charity (Sadaqah), and the one performing the sacrifice includes them in the spiritual reward (Thawab).
Furthermore, slaughtering a single sheep (on one's own) is considered religously better than participating as one of seven people in the sacrifice of a cow or a camel. And Allah the Almighty knows best.
Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.
Does the 'aqīqah count as valid if it is slaughtered before the seventh day from the birth?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The time during which it becomes permissible to slaughter the 'aqīqah begins from the moment the newborn is fully delivered from its mother's womb.
If the animal is slaughtered prior to the birth, it does not count as an 'aqīqah — it is simply considered an ordinary sheep slaughtered for its meat.
And Allah Almighty knows best.
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.