Can I move to another county with my children?

You don't say if you're married and if not, if the father has established his legal paternity. If yes, on either count, you can't move to another country without the approval of the court. If no, then he has no legally established rights and therefore would first have to establish his paternity to have any standing in court to say over where the children live. Whether or not the court would permit you to take the children out of the country depends on a number of criteria including to which country you're taking them. Get an attorney to help you with this, so you're not in the middle of an expensive inter-country law suit.

Yes, as long as there is no court order in place giving him visitation. Once you move, you need to tell him where you are to avoid having the cops how up at your door.