Is "Bismillāhir-Raḥmānir-Raḥīm" a verse (āyah) of Sūrat al-Fātiḥah?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The basmalah ("Bismillāhir-Raḥmānir-Raḥīm") is, without question, a verse (āyah) of the Noble Qur'an, and according to the Shāfi‘ī school, it constitutes a verse of Sūrat al-Fātiḥah itself. Al-Fātiḥah is a pillar of the prayer, and it consists of seven verses, of which the basmalah is one. The prayer is not valid without reciting it, nor if any of the Fātiḥah's verses, letters, or shaddahs (consonant doublings) are omitted or altered.
It is stated in Bushrā al-Karīm (Vol.1/p.202), from the books of the Shāfi‘ī school: "The basmalah is a complete verse of it [i.e., of al-Fātiḥah] in practice — and it suffices, for this to be established, that there be strong probability (ẓann), especially given that this has come very close to certainty (yaqīn), due to the consensus (ijmā‘) of the Companions on its inclusion in the muṣ-ḥaf [the written text of the Qur'an] in its script, despite their extreme diligence in excluding from it anything that did not genuinely belong to it — to the point that they even [carefully regulated] its diacritical dots and vowel markings." And Allah, the Exalted, knows best.
Is the father a Mahram (Non-marriageable) to his son`s mother-in-law?
The father is a non-Mahram (Marriageable) to his son`s mother-in-law, so it is impermissible for them to look at each other, or to have a seclusion (Khalwah).
Should the Zakah (obligatory charity) giver tell the poor recipient that this is the Zakah of his money?
No, he shouldn`t. But, the Zakah giver should make an intention in his heart that it is the Zakah of his money when paying it to the poor. And Allah 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.
