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.
What is the ruling of Islamic Law on eating or drinking during the circumambulation?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
It is disliked (makrūh) to eat or drink during the circumambulation (ṭawāf).
Shaykh al-Islām Imām al-Nawawī states in al-Majmūʿ: "It is disliked to eat or drink during ṭawāf, and the dislikedness of drinking is lighter. Neither of them, nor both together, invalidates the ṭawāf. Al-Shāfiʿī said: 'There is no objection to drinking water during ṭawāf, and I do not consider it sinful; however, I prefer that it be avoided, as refraining from it is more befitting in terms of proper conduct.' Among those who explicitly stated the dislikedness of eating and drinking, and that drinking is the lesser of the two, is the author of al-Ḥāwī."
If, however, a person is in genuine need of drinking, then there is no dislikedness in doing so. In any case, the ṭawāf itself remains valid. And Allah the Almighty knows best.
Is it valid for the 'aqīqah to be performed using the newborn child's own wealth?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is not permissible for the guardian to perform the 'aqīqah using the newborn child's own wealth, because the 'aqīqah is a voluntary act of giving (tabarru'), and a guardian is prohibited from making voluntary expenditures from the child's wealth. Should he do so, he becomes financially liable for what he spent. And Allah Almighty knows best.
What are the conditions for a valid Udhiyah?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
First: The age of the animal must meet the Sharia requirements. These requirements vary depending on the type of sacrifice:
Camels: Must have completed five years and entered their sixth.
Cows: Must have completed two years and entered their third.
Goats: Must have completed two years and entered their third. As for Sheep, they must have completed one year and entered their second.
Some scholars have permitted goats that have completed one year and entered their second.
The Hanafi school, along with an opinion in the Maliki school, permits sacrificing sheep that are at least six months old, provided they are healthy and physically substantial. According to the Shafi’i school, it is permissible if the sheep sheds its front teeth (ajdha') before reaching one year [Al-Iqna’, by Al-Shirbini (Vol.2/P.588)].
Second: Soundness and freedom from defects. The animal must be free from any defect that causes a decrease in its meat or market value. This is based on the hadith narrated by Al-Bara' bin 'Azib, that the Prophet (peace be upon him) said:
"Four [defects] are not permissible in sacrifices: A one-eyed animal whose blindness is evident, a sick animal whose illness is evident, a lame animal whose lameness is evident, and an emaciated animal that has no marrow in its bones." [Reported by Abu Dawood and Al-Tirmidhi, who graded it as authentic].
These defects are detailed as follows:
Evident Lameness: It is not permissible to sacrifice a lame animal if the lameness is severe enough to prevent it from walking to the pasture or seeking food, as this leads to a decrease in its meat. However, slight lameness that does not hinder its grazing is overlooked.
Evident Blindness (One-eyed): It is not permissible to sacrifice a sheep, cow, or camel that has a white film over its eye blocking light, or one that has lost an eye entirely. Weak vision that does not affect its ability to eat does not prevent the sacrifice from being valid.
Evident Illness: An animal with a clear sickness that prevents it from eating or moving is not valid. This includes severe mange (Jarab) that spoils the meat.
Extreme Emaciation: An animal so thin that there is no marrow left in its bones is invalid. The standard for emaciation that invalidates the sacrifice is that which spoils the quality of the meat to the point that people would find it undesirable even in times of plenty.
Additional Considerations:
These are the defects mentioned in the Prophetic tradition, and any defect that causes emaciation or reduces the meat or value is compared to them by analogy. This includes animals that are mentally unstable (diseased), those with mange, or those with a missing ear. In contrast, a slit or pierced ear does not affect the validity of the sacrifice. And Allah the Almighty knows best.