Esther the Queen and Her Son

Hello. It is known that Esther had a son named Darius. I have several questions following what I learned: 1. Was he her son and Ahasuerus' son? 2. It is known that he was the king of Babylon during the Second Temple period and Esther lived with him in Babylon and was already old. I read in the Book of Daniel that Ezra was alive in Babylon and heard about the situation in the Land of Israel and wanted to come here to try to help the Jews. He requested permission from Darius, who was the king. Darius agreed and Ezra went up. After that, Nehemiah also requested permission to go up to the Land for a certain time, and Darius also agreed to him, but it is written in the book that according to his words we understand that he agreed to everyone to go up and do as they wish (something like that) for the Creator, Hashem, so that He would not be angry and punish the royal family. The book states that we see he agreed only because he cared for himself and his family. Additionally, it is written that he consulted with his mother Esther regarding Nehemiah, and she approved, and thus he allowed Nehemiah to go up to the Land. My question is, do I understand correctly that Darius did not act like a Jew? And if so, how was it so? After all, Queen Esther was a righteous person who did good for the Jewish people; how could her son be like this? And how did she even live with him in Babylon? Did she keep the laws of the Torah? I don't quite understand because, again, I know she was a very righteous person... I know the question is not very realistic because, unfortunately, there are Jews today who do not keep the Torah and commandments, but we are talking about Queen Esther and her son... These were not ordinary people.