Is it permissible for a person, who vowed to offer two Rak`ahs (units of prayer) everyday for Allah`s sake, to do so after the Fajr (dawn) prayer?
In principle, a vow should be fulfilled as it was originally intended, and if its time lapses then, it should be fulfilled at any time because of the vow. Therefore, it is permissible to offer them after the Fajr prayer, and there is no prohibition in doing so. However, it is preferable for him to offer these two Rak`ahs at some other time to avoid the disagreement amongst the scholars in this regard.
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. And Allah Knows Best.
Is it permissible for a person who has to make up for missed fasts to fast six days of Shawal (the month directly after Ramadan) before making up for the missed fasts ?
Yes, it is permissible for him/her to do so, but it is better that they make up for missed fasts first. And Allah Knows Best.
During Ramadan, a man had sexual intercourse with his wife. What is the expiation for that? And is he permitted to pay the expiation?
If a man has intercourse with his wife during the daytime in Ramadan, he must make up for that day and offer expiation. The expiation is fasting for two consecutive months. If he is unable to do so, he must feed sixty needy people. And Allah Knows Best.