Nobody's saying "Calling them Roman is not allowed"; I just think that treating them as the heir to the Roman empire - and giving them the cores to suit that status - is kind of silly.What should the Eastern Roman Empire be called then? I'm not just talking EU timeframe but any time frame following the fall of the West. Byzantine was invented long after the fall of Constantinople to make a distinction between the ancient empire and the medieval one. No contemporary sources refered to the Empire in Constantinople as Byzantine, but as either Greek (by the westerners) or Roman (by them themelves). If Roman is for some reason not allowed, then what should it be?
Personally I'd say they stopped being the Eastern Roman Empire around 610-650, when Heraclius and his heirs lost control of most of the former Roman provinces and the Patriarchates of Alexandria, Jerusalem and Antioch, made Greek instead of Latin the official language, changed his title to βασιλεύς instead of imperator, replaced the old Roman-style salaried professional army with feudal levies, abolished most of the old Roman centralised bureaucracy, and virtually stopped minting coins except in Constantinople itself. It wasn't the same state after so radical a change even if they kept the name.
If you want to be politically correct, call it Romania (Ῥωμανία). Otherwise, 'Byzantine Empire' is fine. The word 'Byzantium' was used to describe the empire from the 1500s on, which was only a century after it fell.
In 1200, you might be right. In 1390, not even the Byzantines could be in such denial as not to know who was whose vassal. Look at what happened when the same Emperor, John V, ordered Constantinople's walls to be repaired and strengthened. Sultan Bayezid didn't like that, and in 1391 ordered him to reverse the repairs and even demolish the main gate into the city. John obeyed, like a good vassal.Not necessarily. I don't know about this specific event, but from what I know of the Byzantines in general they were a lot like China in that they considered everyone around them to be 'barbarians only good for giving tribute'. So I'd guess that Byzantine-Turkish alliance the Byzantines considered the Turks to be their vassal who they were merely using to put Philadelphia under their control and that the act of the Byzantines helping the Turks was proof to them that the Turks were only allowed to control anything at the Byzantine's leisure.