Is the aforementioned child considered legitimate, and is it permissible to register him under the father's name? And is the father obligated to remain in Sweden to care for him, or is he entitled to hand over custody to the child's mother, taking into consideration the Prophetic Hadith: "The child is for the bed, and for the fornicator is the stone"?

Your marriage to that woman was invalid because she is neither from the People of the Book nor chaste, and her guardian did not marry her off. As for the child's lineage, if you believed the marriage to be valid, then the child is attributed to you according to jurists like Ibn Taymiyyah, even if the marriage contract was فاسد (irregular/corrupt). You must strive to bring the child to you and raise him upon Islam. You are not obligated to remain in a country where you do not feel safe from temptation. If you can travel with your son, then do so; otherwise, take care of yourself and strive to guide your son from afar. You must also engage in sincere repentance and perform many good deeds.