Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
What is the ruling of Sharia when husband kicks the wife out of his house without a lawful excuse? Moreover, in such case, when she stays at her parents's house for several months, is she allowed to claim maintenance through a Sharia court?
All perfect 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.
It isn`t permissible to kick wife out of house save for a valid reason since Almighty Allah Says (What means): "O ye who believe! Ye are forbidden to inherit women against their will. Nor should ye treat them with harshness, that ye may Take away part of the dower ye have given them,-except where they have been guilty of open lewdness; on the contrary live with them on a footing of kindness and equity. If ye take a dislike to them it may be that ye dislike a thing, and God brings about through it a great deal of good." [An-Nisa`/19]. In addition, it is the duty of the husband to provide for his wife and children in kindness, and this includes food, garment and residence. Moreover, dialogue and supplication are the best means for solving marital problems; however, the wronged party may resort to court. And Allah The Almighty Knows Best.
Is it permissible for a Muslim to escort his deceased disbelieving relative to his final resting place?
It is permissible provided that he doesn`t walk behind non-Islamic symbols, rather he should walk in front of them.
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.
