Are minors(persons under the age of puberty) obliged to fast?
A male under the age of puberty isn`t required to fast, but his guardian should order him to fast if he was able to endure the hardship until he gets used to it.
Is it permissible for a person to sever ties with his sister because her husband had quit prayer and done him harm?
A sister is a blood-relative and maintaining blood-relations is mandatory in Islam. As regards her husband, he should be invited to the way of Allah with wisdom and beautiful preaching.
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 woman who is observing Iddah after her husband`s death to sit with her daughter`s suitor, although their marriage contract hasn`t been concluded yet?
All perfect praise be to Allah the Lord of The Worlds. May His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
A suitor who hasn`t concluded the marriage contract isn`t a Mahram*, so he must be treated as such. And Allah The Almighty Knows Best.
* The period a woman must observe after the death of her husband or after a divorce, during which she may not marry another man.
* In Islam, a mahram is a member of one's family with whom marriage would be considered haram, concealment purdah, or concealment of the body with hijab, is not obligatory; and with whom, if he is an adult male, she may be escorted during a journey, although an escort may not be obligatory.