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.
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.
Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.
What is the ruling on a woman who learns of her divorce or her husband's death after her *ʿiddah* has ended? And does the *ʿiddah* of a woman who abandoned mourning (*iḥdād*) for the whole period of the *ʿiddah*, or part of it, still end, even though she knew that abandoning it is forbidden?**
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
The waiting period of a widow who is not pregnant is four months and ten days, based on the saying of Allah Almighty (which means): “And those of you who die and leave wives behind, they shall wait concerning themselves for four months and ten days” [al-Baqarah/234]. If she is pregnant, her ʿiddah ends with the delivery of her child. The ʿiddah is calculated by lunar months and begins from the date of death.
If her ʿiddah ends before she learns of her husband’s death, or of his divorcing her, she does not observe the ʿiddah upon learning of it, because her ʿiddah has already ended.
It is stated in al-Bayān fī Madhhab al-Imām al-Shāfiʿī (Vol.11/p.41), one of the books of the Shāfiʿī school: “If a man divorces his wife, or dies while he is absent from her, her ʿiddah is from the time of the divorce or from the time of death. If she does not know of the divorce or the death until her ʿiddah period has ended, then her ʿiddah has ended. If she learns of it before the end of the ʿiddah period, she completes her ʿiddah counting from the time of the divorce or death.”
As for a woman who abandoned mourning throughout the ʿiddah while knowing that it is forbidden, she is sinful, although her ʿiddah has nonetheless ended. And Allah Almighty knows best.