1) As for the first story about Imam Hussains wife i am unfamiliar with this.
2) There are others excluding the Imams who are known that the hajjat are sought by them such as Fatima Masooma, Umm al Baneen, Sayed Muhammad in Balad Iraq.
3) Aisha is just a name it doesn’t mean that she is named after that particular Aisha as we see that same thing with children of Imam Ali named Omar. As for the children of Imam Hadi (as) the following is quoted from Kitan al Irshad of Sheikh Mufid
“Abū al-Ḥasan (al-Hādī), peace be upon him, died in Rajab in the year 254 A.H. (868) in his house in Sāmarrā’. Among the children he left behind is Abū Muḥammad al-Ḥassan, his son who was the Imam after him, al-Ḥussein, Muḥammad, Ja‘far and his daughter, ‘Ā’isha. He resided in Sāmarrā’ for ten years and some months until his death. He died at the age of forty-one as we have mentioned before.”
4) Yes Yazid wife was a believer.