Is it permissible to pray in the Umayyad Mosque in Damascus, given the presence of graves within it, and is it true that the Awaited Mahdi will lead Jesus (peace be upon him) and the Muslims in prayer there?

It is permissible to pray in the Umayyad Mosque in Damascus. The Companions and their Successors prayed in it, and no one disapproved of that. What is believed to be the tomb of Yahya (John) - peace be upon him - is not ثابت [proven/established]; rather, it is merely a story with no isnad [chain of narration], and it was not a clearly designated مقام [shrine/station] during the time of the Companions and their Successors.

As for Saladin, he was not buried there; rather, he was buried in his home, and the burial of any contemporary figure therein has not been established. Furthermore, building mosques over graves and incorporating graves into mosques is forbidden by sacred law. As for the descent of Isa (Jesus) - peace be upon him - it will occur at the White Minaret to the east of Damascus, as stated in Sahih Muslim.