Is there any information about Soshandukht or her father?

This is an interesting story, I was not aware of this. Wikipedia makes a source-less claim she was the daughter of Huna bar Natan, who was also an Amoraic sage, mentioned a number of times in the Talmud. This seems strange, because no hint is given anywhere that his daughter was wed to the Persian king. It is clear that he was in good standing with King Yazdegerd, and perhaps they were even friends, but nothing about being connected via marriage.

On this blog in Hebrew, again source-less, it says that she was either the daughter of Kahana bar Abba or his father, Abba bar Ukva. This view is likely based on dating the lives of all of these individuals.

Scholarship over the years has dealt with the story. Naturally, there are disagreements, less about who she was or who her father was and more about whether there really was a Jewish woman married to a Sassanid king. A full list of sources can be found in Simcha Gross's recent comprehensive essay on the subject "The Curious Case of the Jewish Sasanian Queen Šīšīnduxt: Exilarchal Propaganda and Zoroastrians in Tenth- to Eleventh-Century Baghdad", JAOS 141:2, p. 367, ns. 10 and 11. Gross himself opined that the story is a myth created by people connected to the household of the Jewish exilarch (Reish Galuta) of the 10th or 11th century, wishing to strengthen the legitimacy and authority of the exilarch's household and is in line with other traditions and stories that sprouted up around the time.

Some of his points:

• Provincial Capitals is the only extant source that mentions this Shushandukht. There is not even the faintest hint for her existence in any other source, Jewish or otherwise.

• Some elements of the story as brought in Provincial Capitals are obviously fictitious, such as the story which says that the queen settled Jews in the city of Gay, Isfahan. Historical evidence suggests Jews settled in the whole of the province centuries earlier.

• The name Shushandukht literally means "daughter of Susa", which indicates the name is fictitious.

• The story resembles too much the story of Queen Esther: A Jewish girl marries the king of Persia and becomes queen. To this point I will add two more elements: Esther was a descendant of Israel's first royal family, the House of Saul, and according to Jewish tradition, was the mother of Darius I (see for example Esther Rabbah 8:3), in whose time the Second Temple was completed and in general was good to the Jews (Ezra 6:1-15). Likewise, Shushandukht is said to have been from the Jewish pseudo-royal family, that of the exilarchs, themselves purportedly descendants of the House of David (see for example Seder Olam Zuta, 7-10), as well as being the mother of King Bahram, who was good to the Jews.

• Jewish sources that describe the exilarch in the 5th century indicate that though the exilarchs were in good standing with the Sassanid kings, they would not have been considered prominent enough for a king to wed a daughter from that house.

To his points I will add that the story doesn't make sense from a Jewish point of view: It would not have been honorable for a Jewish woman to marry a non-Jewish king, and all the more so the daughter of the exilarch and the sister of the next exilarch, all traditionally descendants of the House of David, and we have no indication from Provincial Capitals or any other source that Yazdegerd converted to Judaism (moreover, Bavli Zevachim 19a seems to indicate the opposite). Likewise, this was and is the position of Jewish tradition towards the marriage of Esther and Achashverosh, the above midrash from Esther Rabbah just one example of many that shows how that marriage was seen.

Jacob Neusner, who also discussed the validity of the story in his book "The History of the Jews in Babylonia, Vol. V" and rejected it, added another similarity between the two stories: Mordechai saved the king's life, and according to Bavli Megillah 13b, Bigtan and Teresh planned on poisoning Achashverosh's drink, and the Amoraic sages in the time of Yazdegerd also saved the king's life from poisoned food (Bavli Ketubot 61a-b). Neusner himself, however, did not think the story of Shushandkht had been based off of the story of Esther. In his view, it was more likely a legend based off of the good relationship between Yazdegerd and the Jews, as well as the actual wives and/or concubines of the king.