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.
Which hour coincides with the last third of the night?
This hour could be figured out by calculating the number of night hours which start at sunset, and end at dawn break, then the total is divided on three to get the result.
Is it permissible to purchase the Udhiyah on installments?
It is permissible to purchase the sacrificial animal (uḍḥiyyah) on installments or by borrowing its price. However, it is not recommended for the poor to do so, because they are not required to offer a sacrifice, and Allah does not burden a soul beyond its capacity. If doing so would lead to negligence in providing for one's dependents (nafaqah wājibah), then giving precedence to obligatory maintenance over borrowing to buy the sacrifice is appropriate. This is based on the saying of the Prophet (peace and blessings be upon him): "It is sufficient sin for a person to withhold food from those whom he is responsible to support."
Furthermore, the sacrifice is valid from one who has a debt, but it is preferable (awlā) to repay the debt first, especially if the debt is due immediately (ḥāll). And Allah Almighty knows best.
What is the ruling on fasting?
Fasting in Ramadan is an individual obligation (Fard ‘Ayn) upon every mature, sane Muslim who is capable of fasting.
Fasting can also be recommended (Mustahabb), such as voluntary fasting on Mondays and Thursdays, fasting on the Day of Arafah for those not performing Hajj, and fasting on Ashura.
Fasting can be prohibited (Haram), such as fasting on the two Eid days, the Day of Doubt (Yawm al-Shakk), and the Days of Tashreeq.
Some types of fasting are disliked (Makruh), such as singling out Friday or Saturday for fasting without a specific reason and fasting on the Day of Arafah for a pilgrim.