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.
Is it a condition that a woman should untie her locks while making Ghusl (ritual bath)?
Ghusl from Janabah (ritual impurity), or menstruation obligates that water reaches the roots of the hair in order for the Ghusl to be valid, but if it doesn`t, then hair locks must be untied for water to reach them, and for Ghusl to become valid.
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]
Does vomiting during the day in Ramadan break the fast?
Intentional vomiting is one of the nullifiers of fasting; whoever vomits deliberately breaks their fast.
However, if vomiting occurs involuntarily, the fast remains valid as long as nothing returns to the body cavity (jauf). If anything is swallowed back, the fast is invalidated.
The Prophet ﷺ said: "Whoever is overcome by vomiting does not have to make up the fast, but whoever induces vomiting deliberately must make it up." [Narrated by Abu Dawood and At-Tirmidhi]