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.
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 meaning of the word aqiqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The 'aqīqah in its linguistic sense refers to the hair upon the head of a newborn at the time of birth.
In its Sharī'ah sense, it refers to the livestock that is slaughtered on behalf of a newborn child.
And Allah Almighty knows best.
Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.