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 a mother giving the Zakat of her wealth to her children?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible for a mother to give her children from the Zakat if they are among those who are eligible for it—such as being poor (Fuqara), possessing no wealth, and not being sufficiently provided for by the maintenance (Nafaqah) of others. This is based on the statement of the Messenger of Allah ﷺ regarding Zaynab, the wife of 'Abdullah ibn Mas'ud (may Allah be pleased with them both): (Your husband and your children are the most deserving of those upon whom you spend in charity) [Narrated by Al-Bukhari].
It is stated in [Al-Hawi al-Kabir, Vol. 8/P.537]: 'As for the wife, it is permissible for her to pay her Zakat to her husband from all the designated shares... Our evidence is the generality of the saying of Allah the Almighty: "Zakat expenditures are only for the poor and for the needy", and the Hadith of Abu Hurairah that the Prophet ﷺ said to Zaynab, the wife of 'Abdullah ibn Mas'ud: (Your husband and your children are the most deserving of those upon whom you spend), and this is taken in its general sense.' And Allah the Exalted knows best.
Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.
What is the ruling on istinja' after urination, and is it done with water and soap or with water only?
Istinja' from urine and stool is obligatory. It is permissible to perform istinja' with water alone, or with toilet paper alone. The best way is with toilet paper then water. If one wishes to suffice with one of them, then water is better. And Allah the Almighty knows best.