Is it permissible for a man to take back his wife after her first divorce without a new marriage contract, knowing that no penetration occurred, and is it permissible for her to marry another man without a new divorce if the resumption of the marriage was invalid, and how should this matter be dealt with when the spouses differ, and is the wife's dislike for her husband considered a divine punishment?

If you divorced her after a valid seclusion (khalwa (definition) sahihah), then the رجعة (revocation of divorce) is valid, because seclusion is considered equivalent to consummation (dukkhul). As for a mere separation, divorce does not occur by it. So, if she is still under your marital bond, she is not permitted to marry anyone else until you divorce her and her waiting period (iddah (definition)) ends. Indeed, it is forbidden to marry a woman who is married or in her waiting period. In case of a dispute, the matter should be referred to the competent authorities, and the custom of delaying the wedding should be taken into account.