Somewhat, it is generally agreed (almost universally) that the Pope of Rome had the Primacy of Honour, being not only the primus, but also the protos. How it is interpreted in terms of jurisdiction and authority is what is in contention. It will be nice if there could somehow implement a complex system of the Petrine Sees vs the Pentarchy, where Chalcedonian Christians fight over which jurisdictional system they want which along with a whole load of other clusterfucks (like the real important ones) will eventually lead to a grand schism
Yes, you are under the correct impression. When the various bishops of the East had disputes, they would often send requests to the Bishop of Rome to arbitrate on their behalf. The modern Ecumenical Patriarch of Constantinople does not really do this unless it is within their jurisdiction (which is incidentally actually quite small).
The thing about Chalcedonian Christianity is that, although both East and the West were often at odd with each other, it didn't get really bad until the Massacre of the Latins and the Sack of Constantinople, et cetera (Fourth Crusade stuff). I'll even go as far to say that the event of 1054 AD wasn't even a Great Schism in that it was just a verbal spat between Cardinal Humbert and Patriarch Ceralius, the former having almost no authority on behalf of the Western Church, since the Pope had died when shit hit the fan. Even after the Fourth Crusade, there were still a sense of "unity" between the Catholics and the Orthodox (not really) in that the Pope even excommunicated the Emperor when he became involved in the Sicilian Vespers.
The latter two paragraphs where the thrust of my argument, tbh. More often than not, politics go in the way of unity as opposed to strictly doctrinal arguments. The only truly doctrinal issue is the Filioque, and in recent years, the Roman and Greek Churches have come to a certain understanding upon the matter (rather interestingly, when the Holy Father concelebrates Mass Ecumenically with a Patriarch or Metropolitan of the Greek Churches, the Nicene Creed is recited without the Filioque).
But, what I was getting at was precisely this idea of Petrine Supremacy, with the three Petrine Sees (Rome, Alexandria and Antioch), those Churches having been established by St Peter, or in the case of Alexandria, St. Mark, who was, as Peter writes in his first epistle, was like a son to him, should be looked on as having an extraordinary jurisdiction. And it was so for centuries in the Church, primarily for the Petrine See of Rome. As early as 95 AD, we have the Bishop of Rome acting outside his immediate jurisdiction. Likewise, when Athanasius, the Patriarch of Alexandria was in dispute with Constantinople, he went to Rome for judgement. A contested jurisdictional primacy of Rome only truly began with Justinian and the elevation of Constantinople to a place of second importance behind Rome for non-Theological reasons. In actuality, Rome had never ascribed to the idea of the Pentarchy in the first place, the Pope having refused to recognize the Quinisext Council.
I guess, in along and roundabout manner, I am generally agreeing with your point about handling Chalcedonian Christianity