Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.
Is a person rewarded for refraining from all that breaks their fast when required to do so, and is it considered a valid fast?
If a person eats or drinks thinking that Fajr has not yet arrived, but then someone informs them that Fajr had already begun, they must refrain from eating for the rest of the day and make up the fast later.
If they observe the required restraint (imsak), they will be rewarded for obeying the command because fulfilling an obligation brings reward.
However, this is not considered a valid fast in terms of rulings. For example:
● It is not disliked (makruh) for them to use a miswak after noon.
● They are not encouraged to hasten the breaking of the fast at sunset.
● Other fasting-related rulings do not apply to them.
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 the ablution of one who touches his wife without a barrier?
In the Shafi'i school, a man's ablution is invalidated by touching his wife if their skins meet (in any place) without a barrier, whether the touch is intentional or accidental. An exception to this is touching hair, teeth, or nails; these do not invalidate ablution. And Allah the Almighty knows best.