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 ruling on undoing braids of hair during the ritual bath?
It is not a condition to undo braids if the water reaches all the hair and penetrates to its base. If the water does not reach except by undoing the braids, then it is obligatory to undo them for the water to reach. And Allah the Almighty knows best.
What is the ruling on making up missed prayers during prohibited times?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to make up (qada’) missed prayers at any time, even during the periods when prayer is generally prohibited. The prayers that are forbidden and considered invalid during these times are 'absolute voluntary prayers' (nafl mutlaq)—which have no specific cause—and voluntary prayers whose cause follows the prayer itself, such as the Sunnah of entering Ihram or the Sunnah of the Istikharah prayer. Furthermore, no prayer is considered disliked (makruh) during these prohibited times when performed within the Meccan Sanctuary (Makkah al-Mukarramah).
It is stated in Bushra al-Karim (Vol.1/P.181), one of the Shafi’i texts: 'It is not forbidden to perform prayers that have a cause that is not delayed (i.e., the cause is preceding), such as making up a missed prayer (fa’itah)—even if it was a voluntary one—and the funeral prayer (janazah); or a cause that is simultaneous, such as the prayer for rain (istisqa’) or the eclipse prayer (kusuf)... and the Sunnah of wudu, the greeting of the mosque (tahiyyat al-masjid), the Sunnah of circumambulation (tawaf), the Sunnah of arrival, and the prostrations of recitation (tilawah) or thankfulness (shukr). These mentioned prayers and their like are not forbidden provided that one does not specifically intend (ta'ammud) to perform them during the disliked time because it is a disliked time. If one does so intentionally, it becomes forbidden, even if it is a mandatory makeup prayer that is due immediately; because in that case, one is acting in defiance of the Sharia. This is in contrast to when one does not specifically seek out that time, even if the prayer happens to fall within it, or if one seeks it for another purpose—such as delaying a funeral prayer to that time so that a larger number of people may pray over the deceased; in such cases, it is permissible and valid... And it is forbidden to perform prayers with no cause at all, like absolute nafl, or those with a delayed cause, such as the Istikharah prayer, the prayer for Ihram, the prayer for a need (hajah), the prayer before leaving the house, or the prayer before execution; because their causes occur after the prayer itself.' And Allah the Exalted knows best."
Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.
