What is the ruling on congratulating others on the arrival of Ramadan?
Congratulating others on the arrival of Ramadan is permissible.
If someone does it to express their love for worship and the blessings of Ramadan, and congratulates their fellow Muslim, they have done something good.
The Prophet ﷺ said: "Shall I not tell you something that, if you do it, you will love one another? Spread peace (Salam) among yourselves." [Narrated by Muslim]
Giving Salam is a prayer for peace and safety.
What is the ruling on mentioning Allah`s name upon slaughtering an animal?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is a Sunnah (prophetic tradition) for the person slaughtering to say: "Bismillah" (In the Name of Allah), though the most complete and perfect phrasing is to say: "Bismillah ar-Rahman ar-Rahim" (In the Name of Allah, the Most Gracious, the Most Merciful). This is in accordance with the words of Allah Almighty: "So eat of that [meat] upon which the name of Allah has been mentioned." (Al-An'am/118)
However, pronouncing it is not an obligatory requirement (Wajib). If someone omits it—whether deliberately or out of forgetfulness—the meat remains entirely lawful (Halal) to eat, though purposely omitting it is considered disliked (Makruh). And Allah the Almighty Knows Best.
What is the waiting period ('Iddah) for a woman whose husband has passed away, and what is the ruling on her wearing gold?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The waiting period ('Iddah) for a woman whose husband has passed away is four months and ten days for one who is not pregnant. As for a pregnant woman, her waiting period lasts until she gives birth. It is obligatory for her to remain in the marital home, only leaving for a necessity. During this time, it is prohibited (Haram) to display any form of adornment on the body or clothing; this includes wearing kohl, gold, all types of perfume, and dyeing the hair. Likewise, it is prohibited to receive a direct marriage proposal or to marry during this period.
It was narrated by Umm 'Atiyyah that the Prophet ﷺ said: 'We were forbidden to mourn for a deceased person for more than three days, except for a husband, for whom the mourning period is four months and ten days. During this time, we were not to use kohl, nor wear perfume, nor wear dyed clothing except for garments made of 'Asb (coarsely dyed yarn). We were, however, granted a concession at the time of purification—when one of us bathed following her menses—to use a small amount of Kust (costus) or Azfar (fragrant substances). We were also forbidden from following funeral processions.' (Narrated by Al-Bukhari). And Allah the Exalted knows best."
Who is the one required to slaughter the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah for the 'aqīqah to be slaughtered by the guardian of the newborn — the one upon whom the child's financial maintenance is obligatory — provided he is financially capable of doing so. 'Alī ibn Abī Ṭālib, may Allah be pleased with him, reported: "The Messenger of Allah ﷺ performed the 'aqīqah for al-Ḥasan with one sheep." — Narrated by al-Tirmidhī. And Allah Almighty knows best.