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.
Is it permissible for a suitor to go out with his fiancé upon the conclusion of the marriage contract?
Upon the conclusion of the marriage contract, the woman becomes lawful for her husband. However, the customary practice should be observed in this regard since rights could be lost in case of divorce, or death.
I want to buy a house by having a loan from a bank?
If taking a loan from usrious banks it is impermissible; on the other hand, if this took place through Islamic ones then it is permissible to get the loan, since Allah the Most Exalted Said (What means): "And Allah has made buying lawful and forbidden usury (riba)." And Allah Knows Best.
An engineer has to work in a circuit system of a non-Islamic bank. What is the ruling on the income she receives from doing this job?
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.
Necessity lies in fearing for one`s life, honor or mind, so the wage received for doing a forbidden job is equally forbidden. Kindly contact the Iftaa` Department to further clarify the nature of your work. And Allah The Almighty Knows Best.