I have a question. I know that the Church condemns us from marrying non-Christians like Jews/Muslims/Atheists/Pagans etc, and that the Church would not marry one of its members to a non-Christian. So, what if, hypothetically, an Orthodox man who was perfectly aware of the Church's stance on this topic got married to a non-Christian girl in a US Court by a Judge, and then later the couple had children. Would the Orthodox Church recognize their marriage as valid and what type of punishment would they give the man?
What do you mean by valid?
The only marriage the Church "recognizes" is that done in church. The penance given to the Orthodox person would depend on the spiritual father's judgment--this all assumes the Orthodox person, who flouted the rules, even wants to be in the Church. If later he brings his children for baptism it presents a quandry. It's not the children's fault, but will they ever be seen in church again?
By 'valid' I mean would he be considered an adulteror/fornicator? And would the Church allow them to continue living together as man and wife since they are at least legally married or would they urge them to separate?
And what if the couple divided the children. Say, the female children were raised under the mother's religion and the male children under the father's religion. And say that the man still does love the Church and wants to remain a part of it even though he willingly broke the rules, could he be exocommunicated or anathematized?
Well, unlike some groups, the Orthodox do not have a blanket solution to every mess. If the Orthodox spouse willfully sins, but repents, it will be dealt with through confession. There may be a penance. We don't pin red A's to anyone. The marriage isn't blessed sacramentally, but the spouse lives up to the ideals of marriage, repents of the sin, and does the best he or she can do, that's all that matters, I think.
As for splitting the children, this is frowned on. If there's a mixed marriage, the children should be raised Orthodox. However, since the rules were flouted in the first case, one does what one can. Say the spouse repents, it would be hoped that he or she has the sense to see that his or her family's spiritual situation is grave. He or she needs counsel from the spiritual father.
Excommunication is a penance. It is not damnation. The person is denied communion for a time, for the sake of repentance, to know the gravity of the sin. Anathemas are for heretics. There may be excommunication, depending on the spiritual father.