Is it permissible for a wife to ask for permission to go out without informing her husband of the reason for her outing, and is it permissible for her to go out if her husband forbids her from doing so?

There is no objection to a wife seeking her husband's permission to go out for a need without informing him of what that need is. The husband has the right to grant or deny her permission. If he does not grant permission, she is not allowed to go out except out of necessity.