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.
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.
Can someone who begins a voluntary fast break it?
It is preferable for someone who begins an act of worship not to break it.
Allah the Exalted has said {what means}: "and let not your [good] deeds come to nought!" [Muhammad/33].
However, if a person starts a voluntary fast (nafl) and needs to break it, they are going against what is preferable, but there is no sin upon them.
Does vomiting during the day in Ramadan break the fast?
Intentional vomiting is one of the nullifiers of fasting; whoever vomits deliberately breaks their fast.
However, if vomiting occurs involuntarily, the fast remains valid as long as nothing returns to the body cavity (jauf). If anything is swallowed back, the fast is invalidated.
The Prophet ﷺ said: "Whoever is overcome by vomiting does not have to make up the fast, but whoever induces vomiting deliberately must make it up." [Narrated by Abu Dawood and At-Tirmidhi]