Is it permissible for someone to make a supplication to Allah to marry an already married, or an engaged woman?
No, it isn`t, and the person who does so is sinful before Allah, The Almighty.
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.
Is ablution invalidated when blood comes out of the nose, or a wound?
Blood coming out of the nose, or a wound does not invalidate ablution, but it is preferable to make ablution as a way out of the scholars disagreement in this regard.
Is it permissible for a Christian man to embrace Islam in secret and is it permissible for a Muslim woman to marry him?
Praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
His embracing Islam in secret is between him and Allah; however, it isn`t permissible for a Muslim woman to marry him except after he declares his Islam and it is ascertained that he is sincere about that. And Allah The Almighty Knows Best.