Is it permissible to offer an Udhiyah on behalf of the deceased?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
Offering an Udhiyah (sacrificial animal) on behalf of a deceased person is permissible. This is the official position of the Hanbali school (as stated in Kashshaf al-Qina’ by al-Bahuti,Vol.6/P.428) and was also upheld by the prominent Shafi'i scholar Al-’Abbadi (mentioned in Bidayat al-Muhtaj by Ibn Qadi Shuhbah,Vol. 4/P.358). It has likewise been narrated as a valid view among some Maliki and Hanafi scholars.
In fact, Imam Abu Dawud dedicated an entire chapter in his Sunan collection entitled, "Chapter on Sacrificing on Behalf of the Deceased." In it, he recorded a narration from Hanash, who said: "I saw 'Ali sacrificing two rams, so I asked him, 'What is this?' He replied, 'The Messenger of Allah (peace and blessings be upon him) commanded me to offer a sacrifice on his behalf, so I am sacrificing on his behalf.'"
Imam Abu Dawud also narrated from Jabir (may Allah be pleased with him) that the Prophet (peace and blessings be upon him) said: "O Allah, this is from You, for You, and on behalf of Muhammad and his Ummah (community). In the Name of Allah, and Allah is the Greatest," and he then slaughtered the animal.
The textual evidence here lies in the fact that our Master, the Prophet (peace and blessings be upon him), offered a sacrifice on behalf of his entire community—and it is well-established that his community includes those who have already passed away.
Furthermore, there is an abundance of sacred texts demonstrating that the rewards of righteous deeds reach the deceased. For instance, it is permissible to fast on behalf of a deceased person who passed away with missed obligatory fasts, and it is equally permissible to perform Hajj on their behalf, both of which are firmly established in authentic Hadiths. Therefore, if the reward of fasting (which is a purely physical act of worship) and Hajj (which is a joint physical and financial act of worship) can reach the deceased, then the reward of an Udhiyah reaches them with greater reason (by way of A Fortiori argument). This is because it is a purely financial act of worship, falling under the general category of charity (Sadaqah).
Additionally, scholars have reached a consensus (Ijma') that the rewards of charity reach the deceased, and since the Udhiyah is inherently an act of charity, it falls under the same ruling. Consequently, based on all the aforementioned evidence, we hold the view that offering a sacrifice on behalf of the deceased is entirely permissible. And Allah the Almighty Knows Best.
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.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
Should a person feel pleased, or have a virtuous vision after offering Istikhara (guidance prayer) in order to do what he/she had offered it for?
The result of the Istikhaarah is not necessarily that a person sees something (in his dream), or feels pleased, but the most important result of the Istikhaarah is whether a person is enabled to do a given matter or not.
