What is the ruling on using eye drops while fasting?
Eye drops do not break the fast, even if the taste reaches the throat, because the eye is not an open passage to the body cavity.
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 on speaking while relieving oneself?
It is disliked for one relieving himself to speak while doing so, except out of necessity. And Allah the Almighty knows best.
If a woman becomes pure from menstruation shortly after Fajr in Ramadan, what is required of her?
If a woman becomes pure (from menstrual period) after Fajr, even shortly after, it is recommended (but not obligatory) for her to refrain from eating and drinking for the rest of the day. However, she must make up for that day after Ramadan.
She will be rewarded for both refraining from eating (imsak) and making up the fast (qada) since she was menstruating for part of the day.