Is this marriage considered Islamically valid, given the wife's refusal of the dower, the condition of divorce upon leaving without permission, and the Imam's ruling that the marriage is void?

It appears from the question that the marriage took place without the woman's guardian (wali), which is invalid according to the majority of scholars, and must be annulled and renewed with the presence of the guardian. Imam Abu Hanifa, however, views this contract as valid if it fulfilled the remaining conditions. Reciting Al-Fatiha is not a condition for the validity of the marriage. If the wife is content with the dowry, she has the right to relinquish it to her husband. If the husband conditioned his wife's divorce on her leaving their home without his permission and she did leave, then the divorce has occurred according to the majority. The husband has the right to take back his wife as long as her waiting period ('iddah) has not expired, unless it was the third divorce.