Is a girl considered legally married after she reconciles with her husband during the 'iddah period without her family's knowledge, and is it permissible for her to move in with the husband despite her father's refusal?

As long as you have not consummated the marriage with your wife, your divorce is considered an irrevocable divorce (talaq bain) according to most scholars. You do not have the right to take her back except through a new marriage contract. If the woman agrees to return to you, her father has no right to prevent her, based on the Almighty's saying: "And when you divorce women and they have fulfilled their term, do not prevent them from remarrying their [former] husbands if they agree among themselves on an honorable basis. That is instructed to whoever among you believes in Allah and the Last Day. That is purer for you and cleaner. And Allah knows, while you know not."

However, the Hanbalis hold that a valid seclusion (khilwah sahihah) takes the place of consummation in making the divorce revocable. According to their view, your taking your wife back is valid, and her father has no right to prevent her. This latter view is the one we adopt for fatwa. The view of the majority (jumhur) is more cautious, and therefore, we advise you to renew the marriage contract.