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 using a miswak (tooth-stick) during the day in Ramadan?
It is permissible to use a miswak before noon while fasting. However, according to the Shafi'i school, it is disliked (makruh) after noon in order to preserve the natural effect of fasting in the mouth, as the Prophetﷺ said in authentic hadith:
"The breath of a fasting person is more pleasant to Allah than the fragrance of musk."
What is the ruling on entering the toilet with something containing the remembrance of Allah?
It is disliked to bring into the toilet anything containing a reminder of Allah, such as the Quran or a sacred name (like the name of Allah, Muhammad, etc.). If one does that, etiquette dictates that he conceal it so it is not visible while entering the toilet, either by cupping his hands over it, putting it in his pocket, or inside his shirt if it is a necklace or ring. And Allah the Almighty knows best.
Is it permissible to appoint a proxy for the sacrificial offering outside Jordan?
In the name of Allah; all praise is due to Allah, and peace and blessings be upon the Messenger of Allah.
It is permissible to appoint a proxy—whether an individual or a charitable organization—to perform the sacrificial slaughter (Udhiyah) on one’s behalf, even if it is carried out in a country other than that of the donor. This is conditional upon the proxy’s adherence to the established requirements of the Udhiyah, including the animal’s age, its freedom from physical defects, the designated timing of the slaughter, and the proper distribution of the meat.
However, it is preferable for the one offering the sacrifice to perform the slaughter personally, in order to attain the full reward and blessings of the act. And Allah (Exalted be He) knows best.