What is the ruling on applying perfume while fasting?
Applying perfume does not break the fast. However, it is better to avoid it, as fasting is a practice of simplicity and restraint, while perfume is a form of luxury.
What is the Islamic ruling on eating from the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is recommended (mandūb) to eat from the 'aqīqah, unless it was vowed (mandhūrah). If the 'aqīqah was made as a vow, then it is prohibited to eat from it, and it becomes obligatory to distribute the entirety of it in charity to the poor. And Allah Almighty knows best.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
Is it permissible for a mother to stop her daughter from marrying a respectable, God-fearing young man because he isn't college/university graduate?
It is impermissible to deny a woman her right in getting married except for a lawful reason, and problems can be solved through dialogue. However, she can file a suit in order for the judge to have a say in that regard.