Does the rebellious wife (nashiz), who receives maintenance for herself and her son from a foreign country, have the right to demand child support from the husband, especially if she exploits the laws of that country to live with the children without the father?

The wife is disobedient (nashizah) due to her travel without her husband's permission, and thus her right to maintenance (nafaqah (definition)) is forfeited. As for the son's maintenance, it is not obligatory for the father if the son has his own wealth, as is the case here where the state sends him money.

The husband must advise his wife and explain to her the impermissibility of her disobedience (nushuz), and that residing in disbelieving countries is not permissible by Islamic law. If the wife insists on her stance, then it is better to divorce her. In the event of divorce, the father has more right to the custody of the son if he is young, especially if the mother resides in another country.