Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
Must a person refrain from eating for the rest of the day if they break a fast of a vow (nadhr) or a make up fast (qada)?
Whoever observes a vowed fast (nadhr) or a makeup fast (qada) is prohibited from breaking it without a valid excuse. If they break it without a legitimate reason, they are sinful.
However, they are not required to refrain from eating for the rest of the day, because such restraint is only required out of respect for the month of Ramadan, not for other types of fasting.
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.
How is Laylat al-Qadr observed?
Laylat al-Qadr is observed through various acts of worship and obedience, including:
● Performing obligatory, Sunnah, and voluntary prayers
● Reciting the Quran
● Increasing remembrance (dhikr) of Allah
● Giving charity
● Performing I‘tikaf
● Inviting others to Allah
It is also Sunnah to increase in supplication and humbly beseeching Him, Glorified and Exalted be He.