Is it obligatory to have the intention for each day of fasting, or is one intention sufficient for the whole month?
The intention is obligatory for each day of Ramadan because each day is an independent act of worship separate from the others.
The intention must be made at night before the break of dawn, as the Prophetﷺ said: "Whoever does not intend fasting at night, there is no fast for him." [An-Nasa’i]
And he also said: "Whoever does not firmly resolve to fast before dawn, there is no fast for him." [At-Tirmidhi, Abu Dawood, and An-Nasa’i]
Whoever wakes up and eats Suhoor while mindful of fasting has made the intention. Likewise, one who firmly intends at any moment during the night to fast the next day has also fulfilled the intention.
What is the ruling on someone who is uncertain whether he omitted one of the ab‘āḍ of the prayer, such as being uncertain whether he omitted the first tashahhud?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The ab‘āḍ include the middle tashahhud and the qunūt in Fajr prayer — for which sujūd as-sahw (the prostration of forgetfulness) is prescribed if left out. Whoever is uncertain, during the prayer, whether he omitted one of these elements, should proceed on the assumption that he did not perform it, and should perform sujūd as-sahw accordingly.
It is stated in ‘Umdat as-Sālik (p. 63): "If he is uncertain... whether he omitted a specific ab‘āḍ element, or whether he performed sujūd as-sahw, or whether he prayed three or four rak‘ahs — he proceeds on the assumption that he did not do it, and performs sujūd as-sahw."
The original principle (al-aṣl) for someone uncertain about [having performed] the first tashahhud is that he assumes he did not perform it. And Allah, the Exalted, knows best.
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.
Is the prayer of someone who didn`t face Quibla (Ka`ba direction) valid?
All perfect praise be to Allah,The Lord of The Worlds. Facing the Quibla is a condition for the validity of prayer, and the prayer of the person who hadn`t faced the direction of the Quibla is invalid, and he is obliged to repeat it. And Allah Knows Best.