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.
Should one who doesn`t perform prayer out of laziness make it up later, and how should he/she do so?
All perfect praise be to Allah,The Lord of The Worlds He/she is obliged to make up missed prayers by offering with each obligatory prayer another one, and if he/she offers two, then it is better. And Allah Knows Best.
Can a woman leading other women in Tarawih prayer recite aloud?
Yes, a woman leading other women in Tarawih prayer is allowed to recite aloud, as long as no non-mahram (other than unmarriageable kin) men can hear her.
What is the ruling on Jumu‘ah (Friday) prayer for a traveler, and what are the conditions for it to be waived from him?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Jumu‘ah (Friday) prayer is not obligatory upon a traveler, whether on a long or short journey, provided that — according to the Shāfi‘ī school — he departs from his place of residence before the time for Fajr prayer begins, and that he does not intend to stay in the place he travels to for four days or more, excluding the day of arrival and the day of departure. If he intends to stay longer than that, he may not make use of the travel concession simply by arriving in that town. If he begins his journey after the adhān for Fajr has already been called, then Jumu‘ah prayer remains obligatory upon him, whether in his own town or in any town where Jumu‘ah is being held.
It is stated in Bushrā al-Karīm (Vol.1/p.383), from the books of the Shāfi‘ī school: "Jumu‘ah [prayer] is not obligatory upon a traveler on a permissible journey, whether long or short, if he departs from his place of residence before Fajr."
A "short journey" refers to travel covering a distance shorter than that of a long journey — that is, less than (81 km). The traveler begins to be entitled to travel concessions once he passes beyond the built-up limits of his town. And Allah, the Exalted, knows best.