Looking back: how does the evolution of the christian world in the third-wave era compare with that of tang and song dynasty china and of the islamic world?

Well we've got the Byzantine Empire and Western Europe (the split between Roman Catholicism and Eastern Orthodox Christianity). Western Europe is a fractured and feudal society while Byzantine lives as a shadow of the former Roman Empire.

Tang and Song Dynasties could be regarded as China's golden age, although I myself am more partial to the Tang Dynasty. Tang was a time of cultural exchange, more trade, art, innovation, and profitable Silk Roads. The Song Dynasty saw the return of Neo-Confucianism and the quite frankly abhorrent practice of women's footbinding. If you don't know what it is, please don't look up pictures.

The Islamic world saw its own golden age with the Abbasid Caliphate and the Baghdad House of Wisdom. Muslims also controlled the near entirety of the peaceful Indian Ocean trade, making them an extremely profitable people.