If a person divorces his wife without touching her, it is clear that the law of iddah and mehr change. But what about the law of divorce? I think, it stays the same: after the divorce, the man and the woman are free to marry another person immediately (as there is no iddah) and they are also free to re-marry each other. This re-marrying may be repeated two times only. Upon the third divorce, the man will not be able to re-marry the same woman until or unless she marries someone else and, unfortunately, gets divorced again. Is this understanding correct?
I believe your understanding is correct. There is however one thing that I must share with you and regarding which I myself am not clear. Mr Moiz Amjad and some other scholars believe that if one divorces his wife for the first or the second time and the iddah period elapses they can remarry. This remarriage will restore all three instances of right to divorce to the husband. This they believe is because the new nikah means that it is not only upon the husband to revoke the divorce but the active and free will of the woman is involved (in the fresh nikah). She could have decided not to marry him again. Whereas in rujoo' (revoking the divorce) she has no say. Therefore, in case of the new marriage and analogically in the case of divorce prior to consummation of marriage, the right to divorce for three times is restored.