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.
Is it permissible for one who sacrifices on behalf of another, with the latter's permission, to eat from the sacrifice?
It is permissible for someone who sacrificed on behalf of another with their permission to eat from it with their permission, and they stand in their place (act as their representative) in distributing it.
It is stated in Nihayat al-Muhtaj ila Sharh al-Minhaj (Vol.8/P.141): "And he—meaning the one sacrificing on his own behalf, provided he has not apostatized—has the right to eat from a voluntary sacrifice and his sacrificial gift; rather, it is recommended. As for an obligatory sacrifice, it is forbidden for him to eat from it, whether it was designated as such initially or as a liability in his dhimmah. And excluded by what has preceded is if he sacrifices on behalf of another, or if he apostatized, then it is not permissible for him to eat from it, just as it is absolutely impermissible to feed a disbeliever from it."(The text I found indicates the impermissibility of eating) ???
And it is stated in Hashiyat al-Jamal ‘ala Sharh al-Manhaj (Vol.5/P.262): "If a person offers a sacrifice on behalf of a living person with their permission, does he act as their representative in distributing it—on the grounds that permission to sacrifice is permission to distribute it—or does it depend on their explicit permission? This requires consideration, but the first view is not far-fetched." And Allah the Almighty knows best.
What is the ruling on combining (jam‘) and shortening (qaṣr) prayers when travel is repeated on a daily basis?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
A traveler is permitted to avail himself of the concessions of shortening and combining prayers once the distance of his journey reaches (81 km) or more. This is on condition that his journey is a permissible one, and that he begins shortening and combining only after setting out and passing beyond the built-up limits of the city in which he resides, or beyond the administrative boundaries of the area where he lives.
Accordingly, if the distance between one's workplace and one's place of residence reaches the distance upon which the concession of combining and shortening depends, then combining and shortening are permissible once he leaves the built-up area, even if this travel is repeated on a daily basis.
Ash-Shirbīnī (raḥimahullāh) said: "It is permissible to combine Ẓuhr and ‘Aṣr, either as an early combination (taqdīm) in the time of the first, or as a delayed combination (ta'khīr) in the time of the second; and likewise Maghrib and ‘Ishā' — either early, in the time of the first, or delayed, in the time of the second — during a long, permissible journey, in following [the Prophet's practice]. As for the delayed combination, it is established in the two Ṣaḥīḥs from the ḥadīth of Anas and Ibn ‘Umar (raḍiyallāhu ‘anhum). As for the early combination, Ibn Ḥibbān and al-Bayhaqī graded it ṣaḥīḥ from the ḥadīth of Mu‘ādh, and at-Tirmidhī graded it ḥasan." [Quoted with slight abridgment from Mughnī al-Muḥtāj, Vol.1/p.529]
The conditions for combining and shortening have already been explained in Fatwa No. 14 and Fatwa No. 842. And Allah, the Exalted, knows best.
What is the ruling on bypassing a nearby mosque for a distant one?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
If a distant mosque has a larger congregation (Jama‘ah) while the congregation in the nearby mosque is smaller, and the congregation in the nearby mosque is not adversely affected by the worshiper's absence, then it is better for him to pray in the distant mosque with the larger congregation. However, if the congregation in the nearby mosque would be affected by his absence—such as if he is its Imam, or if his presence encourages others to attend—then praying in the nearby mosque is better. This ensures that the congregation is established in two different locations within the community. This is based on the saying of the Prophet ﷺ: 'A man's prayer offered with another man is purer than his prayer which he offers alone, and his prayer with two men is purer than his prayer with one and if they are more (in number), it is more beloved to Allah, the Mighty the Majestic' (Narrated by Ahmad, Abu Dawood, and An-Nasa'i). And Allah the Exalted knows best.