I used artificial nails on my wedding night, but I was surprised to find out that they require at least a week to remove. However, I performed all my prayers during that period. What is the ruling on those prayers?
If these nails prevent water from reaching one's skin then your prayer are invalid and obliged to redo all of them. And Allah Knows Best.
Is it permissible to listen to chants praising the Prophet if accompanied by musical instruments?
If singing, or chanting in praise of the Prophet was accompanied by a forbidden musical instrument, then it is forbidden.
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.
Allah The Almighty Made marrying up to four women lawful, but human beings prohibit that with their own land laws. What is the position of Sharia on this?
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.
What Allah has made lawful, human beings can`t make it unlawful. However, taking the decision to marry up to four women requires material and physical abilities, in addition to having the courage to make such a decision. And Allah The Almighty Knows Best.