What is the status of a conditional divorce if I doubted my intention regarding the condition "opening the door," and has the divorce actually occurred due to my wife opening the door to take out the trash, given that I doubted the matter after her action? What are the limits of permission I can grant my wife to open the door in my absence?

The majority of scholars hold that a conditional divorce occurs if the condition is met, and this is the prevailing legal opinion. However, Sheikh al-Islam Ibn Taymiyyah is of the view that divorce does not occur if the intention behind the oath was to prevent or encourage, rather than to make the divorce conditional, and an expiation (kaffarah) becomes obligatory.

Your statement "we will leave each other" is an allusion to divorce, and divorce does not occur by it unless intended. If you did not intend divorce by this phrase, then no divorce has occurred to your wife, even if the condition was met. However, if you did intend divorce, then her divorce occurs if she does what you made it conditional upon.

Intention is the determining factor in defining the condition. If you intended something specific with your oath, the oath applies to it. If the intention is absent, then reference is made to the reason for the oath. Doubt in intention is like its absence. So, if you doubt your intention, reference is made to the reason for the oath. Since the reason for your oath was to prevent your wife from opening the door to anyone without your permission, no divorce occurred when she opened the door to take out the trash. You may permit her to open the door to a specific person or an unspecified person for a certain period without divorce occurring.