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 public street water if it gets on a person's clothes or body?
The default ruling is that public street water is pure (tahir). If one is certain of its impurity (najasa), then a small amount that gets on the lower part of a person's clothes or body is overlooked (excused). And Allah the Almighty knows best.
What is required of one who doubts the number of rakʿāt during prayer?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
If a person doubts the number of rakʿāt he has prayed, he should build upon the lesser number, as that is what he is certain of. He should then perform the prostration of forgetfulness (sujūd al-sahw) before the final salām at the end of the prayer. ʿAṭāʾ ibn Yasār narrated that the Messenger of Allah ﷺ said: "When any one of you is in doubt during his prayer and does not know whether he has prayed three rakʿāt or four, let him pray one more rakʿah and then perform two prostrations while seated before the salām. If the rakʿah he prayed was a fifth, these two prostrations will make it even; and if it was the fourth, then the two prostrations serve as a humiliation for the devil." (Reported by Abū Dāwūd.)
It is stated in al-Muqaddimah al-Ḥaḍramiyyah — one of the foundational texts of the Shāfiʿī school: "If one doubts whether he has prayed three rakʿāt or four, he is obliged to build upon the lesser number."
However, if such doubt recurs repeatedly and reaches the level of obsessive whispering (waswasah), he should not build upon the lesser number in that case — rather, he should build upon the greater number. And Allah the Almighty knows best.
What is the ruling on applying perfume while fasting?
Applying perfume does not break the fast. However, it is better to avoid it, as fasting is a practice of simplicity and restraint, while perfume is a form of luxury.