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 Islamic ruling on smearing the head of the newborn with the blood of the `Aqiqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is disliked (makrūh) to smear the head of the newborn with the blood of the 'aqīqah, as this is a practice that belongs to the customs of the pre-Islamic era of ignorance (Jāhiliyyah). What is Sunnah instead is to anoint the newborn's head with saffron or another pleasant fragrance. And Allah 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."
What is the ruling on performing Tarawih in sets of four rak‘ahs with only one Tasleem?
It is not permissible to combine more than two rak‘ahs with a single Tasleem in Tarawih prayer. Whoever does so, their prayer is invalid. Shaykh al-Islam Imam Muhammad al-Ramli (may Allah have mercy on him) stated: "If a person prays four rak‘ahs of the Tarawih prayer with a single Tasleem, it is not valid if they did so intentionally and with knowledge [of the prohibition]. Otherwise, it becomes an 'absolute voluntary prayer' (nafl mutlaq); because it is contrary to what has been prescribed." [Nihayat al-Muhtaj, Vol.2/P.127].