Does divorce take place with the statement, "If you leave the house today, you are divorced by two [talaqs]," when the wife did not leave on the same day, and does divorce take place if she leaves on the second day?

If you made your wife's divorce conditional on her going out on a specific day, then her going out on another day does not break the condition. Divorce does not occur through doubt, and mere doubt or imagination that she went out without your knowledge does not cause the divorce to occur, because the original state is the continuation of the marriage. We advise you to avoid making divorce conditional.