Is he the one who permitted the construction of the Second Temple? Did she have only one son with the king?
Who was Queen Ester's son with Achashverosh?
There are traditions that Esther's son was Darius (Tosafoth RH 3b, DH shenath, Leviticus Rabbah 13:5). This would mean that Darius, as the son of a Jewish woman, was a Jew. However this is difficult to reconcile with the Talmud's treatment of his actions (vis a vis his intentions in the bringing of sacrifices) through the lens of identifying him as a gentile (RH 3b).
R. Jacob Hagiz (Halakhoth Qetanoth 2:240) suggests that insofar as he was raised as a gentile, he is treated as a mumar (apostate). And that a mumar, in the Talmud's case here, is treated as a gentile. In which case, though he was as the son of Esther a Jew, he is nevertheless treated as a gentile.
Is he the one who permitted the construction of the Second Temple?
Yes, according to this tradition, it is the same Darius.
Did she have only one son with the king?
I do not believe there is any known evidence of any additional children.