Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
What is the ruling of Islamic Law on selling gold or silver in installments or for a differed price?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is prohibited (haram) to sell gold or silver in installments or for a deferred price; rather, immediate hand-to-hand exchange (Taqabud) is mandatory, otherwise, it is considered usury (Riba). The Messenger of Allah ﷺ said: 'Gold for gold, silver for silver, wheat for wheat, barley for barley, dates for dates, and salt for salt—like for like, hand to hand. Whoever increases or asks for an increase has engaged in Riba; the receiver and the giver are the same in this regard.' (Narrated by Muslim). Furthermore, when exchanging new gold for used gold, they must be of equal weight, or it falls into Riba.
The permissible solution (al-makhraj) is for the merchant to purchase the used gold for cash first, and then sell the new gold for cash in a separate transaction. However, the price must be paid during the sitting of the contract (Majlis al-Aqd) for both deals. Alternatively, the jeweler may take the used gold with the intent of remodeling or repairing it, and then charge a fee for the craftsmanship or repair work. And Allah the Exalted knows best."
What are the conditions that must be met for the 'aqīqah to be valid?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The 'aqīqah is like the recommended uḍḥiyyah in terms of the type of animal, its age, and its required characteristics — since it is a recommended slaughter and therefore resembles the uḍḥiyyah in its rulings.
It is agreed upon without any difference of opinion that the 'aqīqah is not valid with any animal other than livestock (al-na'am — camels, cattle, and sheep). Likewise, an animal with a disqualifying defect — such as obvious illness, lameness, or blindness in one eye — does not suffice, as has previously been explained in the rulings of the uḍḥiyyah. And Allah Almighty knows best.
Is the one offering the sacrifice liable if its meat spoils?
Praise be to Allah, and prayers and peace be upon our Master the Messenger of Allah.
If the meat (of the sacrifice) spoils due to the negligence of the one offering the sacrifice in preserving it, or due to improper storage, then he is liable to compensate for the portion due to the poor (which is estimated as half a kilogram of meat). If the sacrifice was a vowed (mandatory) one, then he is liable for all of it.
However, if he was not negligent, then there is no liability upon him, because its ruling is the ruling of a trust (like an item left in someone's care). And Allah Almighty knows best.