Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
Does Laylat al-Qadr move between the nights of Ramadan?
Laylat al-Qadr is definitely in Ramadan, and it is most likely in the last ten nights, especially on the odd-numbered nights.
For this reason, the Prophet ﷺ commanded observing worship in the last ten nights, and he himself used to dedicate those nights to worship. Many people put special effort into the 27th night of Ramadan, and they are rewarded in any case, whether they actually coincide with Laylat al-Qadr or not, because worshiping on any night of Ramadan carries great rewards.
Is the prayer of a woman who is led by an Imam via T.V. permissible?
All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions. Amongst the conditions for the validity of prayer is that both the Imam and those led by him must be in the same place. Thus, the prayer of the person who is being led by a televised Imam is invalid unless he/she was with him in the same mosque. And Allah Knows Best.
What is the ruling on fasting for those with diabetes, heart disease, high blood pressure, kidney disease, or ulcers?
● A sick person who is completely unable to fast is exempted from fasting and must offer fidyah (feeding a needy person), as Allah Almighty says {what means}: "and [in such cases] it is incumbent upon those who can afford it to make sacrifice by feeding a needy person." [Al-Baqarah/184]. They are not required to make up for the missed fasts.
● A sick person who can fast on some days but not others should fast when able and make up the missed days after Ramadan when possible. No fidyah is required in this case.
● If fasting during the long, hot summer days is too difficult for a sick person, but they can make up the fasts during the shorter, cooler winter days, they should break their fast and make up for it when they are able, without fidyah.