Is a wife obligated to seek her husband's permission every time she leaves the house, including for necessary outings, and what is the ruling regarding a general permission for her to go out?

Most scholars hold that if a husband makes his wife's divorce conditional upon an event, he cannot retract it. If the condition is met, the wife is divorced, regardless of whether he intended divorce or mere threat. Some scholars, like Ibn Taymiyyah, are of the view that if he intended to effect a divorce, he may retract it, but if he only intended a threat, no divorce occurs; instead, he is liable for an expiation for a broken oath (kaffarat yamin).

It is permissible to follow the opinion of those who say that divorce does not occur and consider it an oath. The determinant for what the condition applies to is the intention. If there is no intention, then the reason for the oath is the determinant.

If you give your wife permission to go out, there is no breach of an oath, so no expiation is required of you, and no divorce occurs. General permission to go out is permissible.