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 the multiplication of reward for prayer in the sanctuaries of Makkah and Madīnah limited to the obligatory prayer, or does it include the voluntary prayer as well?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
There is no difference between the obligatory (farḍ) and the voluntary (nafl) prayer in the ḥadīth on the multiplication of reward in the two Sanctuaries (al-Ḥaramayn), because the authentic ḥadīths are unrestricted and general in wording. It is therefore not permissible to restrict or specify them without evidence. Shaykh al-Islām Imam al-Nawawī (may Allah have mercy on him) said: “Our school holds that the superiority of prayer in the two mosques of Makkah and Madīnah is not limited to the obligatory prayer, but includes both the obligatory and the voluntary.” [al-Majmūʿ Sharḥ al-Muhadhdhab, Vol.8/p.476]
The multiplication of reward applies to whoever prays in al-Masjid al-Ḥarām or anywhere within the sanctuary (ḥaram). Al-Shirwānī (may Allah have mercy on him) said: “The more correct view according to al-Nawawī is that the multiplication of the prayer’s reward covers the whole sanctuary, and is not limited to the mosque, nor to Makkah itself.” [Tuḥfat al-Muḥtāj with the glosses of al-Shirwānī and al-ʿAbbādī, Vol.10/p.95]. And Allah Almighty knows best.
What advice do you have for those who spend their nights and days of Ramadan watching TV?
Time is too precious to be wasted on entertainment, let alone in the blessed month of Ramadan, where a Muslim should strive for reward and forgiveness.
Is Zakah (obligatory charity) due on the Zakah money received by a poor person, and reached a Nissab (minimum amount liable for Zakah), and a whole lunar year had lapsed over having it in his possession?
Yes, the poor who possessed a Nissab for a whole lunar year is obliged to pay the Zakah due on that money even if it was given to him as a Zakah money in the first place. And Allah Knows Best.