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.
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 to give lessons to mixed students of the seventh grade?
It is impermissible to give private lessons to mixed students since such an act leads to serious consequences for the students in this grade are teenagers.
What is the ruling on someone who vowed to fast for two consecutive months, but failed to do so?
He is obliged to fulfill his vow by fasting for two consecutive months, but if he fails to do so then, it remains a debt on him until he either fulfills his vow, or dies. In case of his death, his guardian is supposed to fast on his behalf, or donate food from his estate in return for fasting, but if the guardian fails to do so then, it is all in the Hands of Allah.