Is it permissible for a woman to go to the market to buy household necessities when there is a man who offered to go, and is he allowed to prevent her if she insists, and is it a condition for her going out that the responsible man not be present?

It is better for a man to go to the market than for a woman, because the basic principle for a woman is to remain in her home unless there is a necessity or a need. She should not leave her home—even for a good deed—without her husband's permission. The conditions for her permissible departure are: that she does not leave except with her husband's permission, that her departure does not lead to neglecting what Allah has entrusted her with, and that she leaves with full modesty, observing the manners of the market.