Is a wife considered disobedient (nashiz) and rebellious if she leaves the marital home without her husband's permission, refuses to return and care for her children, while also communicating with non-mahram men, and should she be divorced?

If the situation is as you described, that you appeased her and she became satisfied, then she is considered disobedient (nashizah) for leaving your house without your permission. Disobedience occurs when she leaves her husband's home without his permission, unless it is out of necessity. Remind her that she accepted your apology and agreed to stay, and that you both moved past the disagreement. It is not permissible for her to speak with non-mahram men in a way that leads to temptation (fitnah). Seek mediation from wise individuals among her relatives. If she responds positively, then praise be to Allah. Otherwise, you are permitted to divorce her.