When does the time for the 'aqīqah lapse and pass?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
If the guardian was financially capable (mūsir) during the majority of the postnatal period (nifās) — which is sixty days — then the obligation of the 'aqīqah remains upon him until the child reaches the age of maturity (bulūgh). Once the child reaches maturity, the demand falls away from the father and those like him. At that point, it becomes Sunnah for the child himself to perform the 'aqīqah on his own behalf.
However, if the guardian was financially incapable (mu'sir) during the postnatal period, and then became financially capable after its expiry — that is, after sixty days — the 'aqīqah is no longer required of him. And Allah Almighty knows best.
What should one do if they see someone deliberately eating or drinking openly during Ramadan?
They must enjoin what is right and forbid what is wrong. If they fear harm from that person, they should disapprove of it in their heart but avoid sitting with them if possible. It is also preferable to seek the help of the authorities to prevent them from doing so.
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 sending blessings and peace upon the prophet (PBUH) after the Adhan?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is a Sunnah to send blessings and peace (As-Salat wa al-Salam) upon the Prophet ﷺ after the Adhan and the Iqamah, for both the caller (Mu’adhin) and the listener. It is permissible to do so aloud after the conclusion of the Adhan. However, it is preferable to maintain a brief pause between the words of the Adhan and the prayers upon the Prophet ﷺ, so that people do not mistakenly assume that sending blessings upon him is an integral part of the Adhan itself. And Allah the Exalted knows best.