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 person to give the Zakah (obligatory charity) to his daughter-in-law?
Yes, it is permissible for a person to give the Zakah to his daughter-in-law if she was poor, and none provided for particularly by whom are obliged to provide for her. And Allah Knows Best.
When is a child instructed to fast?
A child is instructed to fast at the age of seven if they are capable of fasting and have reached the age of discernment, by analogy to prayer. They should be encouraged but not forced, so they can become accustomed to it. It is obligatory for their guardian, whether a father or another, to instruct them.
What is the ruling of Islamic Law on speaking with one's fiancée (before the marriage contract) by telephone and WhatsApp?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Before the marriage contract, the two engaged persons remain non-mahram (strangers) to one another. Reciting al-Fātiḥah, receiving some money on account of the dowry (mahr), or accepting a gift are all merely a promise of marriage and not a valid contract under the Sharīʿah.
It is therefore forbidden for the engaged couple, before the marriage contract is concluded, to be alone together (khalwah) or to shake hands. It is likewise forbidden to speak by telephone or correspond without need. And Allah Almighty knows best.