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.
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 should the one performing the slaughter say at the time of slaughtering the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah to say the following supplication at the time of slaughtering the 'aqīqah:
"Bismillāhi wallāhu Akbar, Allāhumma laka wa ilayka, hādhihi 'aqīqatu fulān."
("In the name of Allah, and Allah is the Greatest. O Allah, this is for You and unto You — this is the 'aqīqah of [the child's name].")
This is based on what al-Bayhaqī narrated in Al-Sunan al-Kubrā (Vol.9/P.511), that the Prophet ﷺ said: "Slaughter it in his name and say: In the name of Allah, and Allah is the Greatest. O Allah, this is for You and unto You — this is the 'aqīqah of so-and-so."
And Allah Almighty knows best.
What is the virtue of performing ‘Umrah in Ramadan?
Ibn ‘Abbas narrated: "When the Prophet ﷺ returned from his Hajj, he said to Umm Sinan Al-Ansariyyah: ‘What prevented you from performing Hajj?’ She replied: ‘Abu So-and-so (referring to her husband) had two camels—he performed Hajj on one of them, and the other was used to irrigate our land.’ The Prophet ﷺ then said: ‘Performing ‘Umrah in Ramadan is equivalent to Hajj with me.’" [Narrated by Al-Bukhari]
The Prophet ﷺ also said: "An ‘Umrah in Ramadan is equivalent to a Hajj." [Narrated by Al-Tirmidhi]
For those who miss the opportunity to perform ‘Umrah in Ramadan, there are many other ways to earn great rewards. One of them is praying Fajr in congregation, then remaining in the mosque remembering Allah until sunrise, and praying two rak‘ahs.
The Prophet ﷺ said: "Whoever prays Fajr in congregation, then sits remembering Allah until the sun rises, and then prays two rak‘ahs, will receive the reward of a complete Hajj and ‘Umrah—complete, complete, complete." [Narrated by Al-Tirmidhi]