What is the semi-circular structure attached to the Holy Kabah?
The semi-circular structure attached to the Holy Kabah is Hijr Al-Kabah (is a low wall originally part of the Kabah) and passing behind it is an obligation while making Tawaf because it is a part of the Holy Kabah. And Allah Knows Best.
My husband told me that he concluded our marriage with a fake name that belongs to another person, because he was sentenced. Nowadays, he recieved an ID, passport and birth certificate with the his current name. What is the ruling on being married to him?
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. As regards being married to the person mentioned in your question: The marriage is valid so long as he was the same person your guardian concluded the marriage with, even if he changed his name i.e. if your marriage contract was concluded with the same person, since what counts regarding marriage is the persons not names. And Allah Knows Best.
Is it permissible for a woman to visit the doctor while in her `Iddah (waiting period)?
It is permissible for her to do that during day time, but it is impermissible for her to leave her house at night except for a sound reason. And Allah Knows Best.
Should Al-Basmalah (i.e. saying Bismillaaahi Ar-Rahmaani Ar-Raheemi) be recited after Takbeerat Al-Ihram (saying Allahuakbar) which marks the beginning of prayer?
All perfect praise be to Allah,The Lord of The Worlds A praying person is obliged to recite Al-Basmalah at the beginning of Chapter Al-Faatihah, and in every Rak`ah (unit of prayer) because it is a verse of it. And Allah Knows Best.