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.
I vowed to give a specific charity if a certain matter came to pass — what is the ruling on giving that charity before the matter is realised?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
Fulfilling a vow (nadhr) is obligatory, in accordance with the word of Allah the Almighty: "And let them fulfil their vows." [Al-Ḥajj/ 29] And the saying of our master the Messenger of Allah ﷺ: "Whoever vows to obey Allah, let him obey Him; and whoever vows to disobey Him, let him not disobey Him." (Reported by al-Bukhārī.)
The Shāfiʿī scholars distinguished between a financial vow (nadhr mālī) and a bodily vow (nadhr badanī). They permitted the fulfilment of a financial vow to be brought forward — before the stipulated condition is met — but did not permit the same for a bodily vow, which may only be fulfilled after the condition has actually been realised.
Shaykh al-Islām Imām Zakariyyā al-Anṣārī, may Allah have mercy upon him, states: "It is permissible to bring forward the fulfilment of a financial vow before the condition stipulated in it is met — such as saying: 'If I am healed, I vow to free a slave' or 'to give such-and-such in charity' — just as it is permissible to pay zakāh in advance. This is unlike a bodily vow, such as fasting." [Asnā al-Maṭālib, vol. 4/P.246]
Imām al-Bājūrī, may Allah have mercy upon him, states: "Like expiation other than fasting, a financial vow — such as saying: 'If Allah heals my sick one, I vow to free a slave for the sake of Allah,' or 'If Allah heals my sick one, I vow to free a slave on the Friday following the recovery' — it is permissible to bring it forward before the recovery in the first case, and before the Friday following the recovery in the second case." [Ḥāshiyat al-Bājūrī ʿalā Sharḥ Ibn Qāsim, Vol.2/P.596] And Allah the Almighty knows best.
Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. And Allah Knows Best.
What is the best manner in distributing the meat of the Aqeeqah?
It is better to divide all of it as cooked meat amongst the poor and the needy. Dividing it into three portions is a Sunnah, but it is permissible to keep all of it provided that a portion, even a small one, is given to the poor and needy (one kilo for example).