What is the semi-circular structure abutting the Holy Kabah?
It is the Black Stone of the Holy Kabah, and one should pass from behind it while offering Tawaf(circumambulation) because it is part of the Kabah.
Is it permissible for the doctor to uncover the hair of a female patient out of necessity?
It is only permissible for him to do so when there is no female doctor, and he should examine the patient in the presence of one of her Mahrams(unmarriageable kin).
Is a woman`s nephew by suckling considered a Mahram( i.e. unmarriageable)who is permitted to accompany her to Hajj?
All that is prohibited by lineage is prohibited by suckling, and a nephew by suckling is a Mahram like a nephew by lineage ;therefore, it is permissible for him to be a Mahram for his aunt in Hajj and Umrah.
I have vowed that if Allah gave me what I wanted so bad, I would fast every Monday and Thursday to the rest of my life; unless there is a valid excuse hindering me from doing so. Fortunately, Allah gave me what I wanted, so I started fasting Mondays and Thursdays; however, I stopped out of laziness. What is the Sharia ruling on this?
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Allah The Almighty described the righteous people of Jannah where He Said (What means): "They perform (their) vows, and they fear a Day whose evil flies far and wide." [Al-Insan/7]. Accordingly, you must do your best to fulfill your vow. However, if you are unable to do so, then feed a poor person for every missed day of fast (About half a JD), and if you are unable to do that, then offer expiation for breaking a vow, which is feeding ten poor Muslims (About half a JD for each), ask Allah for forgiveness and increase remembrance of Him. And Allah The Almighty Knows Best.