I`m infatuated with a man, is it permissible for me to make supplication that he falls in love with me and becomes my husband?
Islam has honored women by being proposed to by men, and not the other way around, so it is inappropriate for a woman to propose to a man since one who hastens in asking for a thing prematurely shall be punished by deprivation. In fact, such a phenomenon is the result of unlawful mixing between the two sexes.
Is it permissible for a wife to leave her house without asking her husband?
It is impermissible for the wife to leave her house without asking her husband except for a sound reason.
A young man who works for a conventional bank has proposed to me. Is it forbidden to marry him because he works there?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon Prophet Mohammad and upon all his family and companions.
Since the Messenger of Allah (PBUH) said: "When someone with whose religion and character you are satisfied asks your daughter in marriage, accede to his request. If you do not do so there will be temptation in the earth and extensive corruption."
Marrying such a person is permissible because he is a Muslim, and Allah may bless him with another job in which no usury/interest is involved. And Allah The Most Exalted Knows Best.
Is prayer mandatory on a dying person who goes through a state of unconsciousness, then wakes up?
If he wakes up and was conscious, then he should pray the way/manner he could, but if he wasn't, then he is exempted from obligatory prayer as Allah Says in this regard (What means): "On no soul doth God Place a burden greater than it can bear." [Al-Baqarah/286].