Is it permissible to trim the eyebrows if they become excessively long?
Trimming eyebrows is permissible if they become excessively long, yet; it is reprehensible. And Allah Knows Best.
Where should a woman following her husband or another man in prayer stand?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
If a single woman prays with a man—whether she is his wife or a non-mahram (Ajnabiyyah)—the Sunnah is for her to stand behind the Imam and be shielded by him. If the Imam is leading both a man and a woman, the man should stand to the Imam's right, and the woman should stand behind the male follower (Muqtadi). In this way, she is shielded from the Imam by the male follower and remains at a distance from the follower and his line of sight. And Allah the Exalted 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.
How is the meat of the 'aqīqah to be distributed?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
If the 'aqīqah is a recommended one (mandūbah), it is obligatory to give some portion of it in charity to the poor — even if the amount is small — with the minimum being approximately half a kilogram of meat.
The most preferable manner of distribution is to divide it into three equal portions, as with the uḍḥiyyah:
One third for the guardian and his household to eat from
One third to be given in charity to the poor
One third to be gifted to friends and neighbours, even if they are wealthy
It is furthermore more preferable to send the food to the poor already cooked, rather than inviting them to come and eat. And Allah Almighty knows best.