Well prior to the Great Schism and the evolution of the Papacy Rome was counted as one of the 5 Patriarchates along with Constantinople, Alexandria, Antioch, and Jerusalem. Prevent the Schism or butterfly some details and you can have the Pope continue to use the title Patriarch. Western Emperors is also pretty easy, the Papacy was frequently held be scions of various noble dynasties especially as the Middles Ages wore on and some of the German noble houses were very big, with lots of cadet branches. Eastern Emperor is more difficult, though if the Latin Emperor counts then that shouldn't be too hard either, it passed through a number of Western noble families before being crushed so if we put it into the hands of our Germanic family then we're looking good.
But the first and the last look impossible Caliph is an explicitly Muslim title and I can't see how that can be combined with Pope while Khagan seems equally impossible.