Is a divorce valid if it occurred via a text message stating: "By God, it is over between us. You are divorced"? What is the ruling on the husband denying it despite the existence of proof? Does the wife become unlawful to him, and is she required to have a new engagement and marriage contract after the 'iddah period ends?

Divorce occurs through writing if the husband intends it. If the wife claims divorce and the husband denies it, his word is accepted unless she provides evidence. If the wife keeps a written message in which the husband explicitly declared divorce, that is sufficient to prove the divorce has occurred. If this is the first or second divorce, the husband may take back his wife as long as she is within her waiting period (iddah (definition)). If the waiting period expires and he has not taken her back, she does not become lawful for him except through a new contract that fulfills its conditions.