What is the ruling on a man remaining with his divorced wife during her 'iddah period after he authorized her to divorce herself, knowing that his madhhab considers embracing and the like as an agreement to take her back, and then marrying her after the 'iddah period ends?

As long as you delegated to your wife the right to divorce herself, and you did not revoke that delegation, and you did not have sexual intercourse with her until she divorced herself, then the divorce has occurred. And with the completion of three menstrual cycles after her divorce, the waiting period (iddah (definition)) ended, and she became irrevocably separated from you. You did not have the right to take her back except with a new marriage contract. Her sleeping beside you after the expiration of the waiting period is impermissible (haram (definition)). And if she married after that, her marriage is valid.