1) I have not seen any hadith in our sources that al-Aqsa mosque will be destroyed and a third temple being built in its place.
2) As for Dajjal, we have very little hadiths about him in Shia literature. We have a hadith that states he will exist until the Mahdi (a) reappears, and then when Jesus (a) comes down he will kill the Dajjal. Another hadith states that Imam Mahdi (a) will kill the Dajjal. So we can conclude that they’ll both kill him