Salve, Sempronii Formose -
Good points, I think. I think that even if one can delineate an end point for the Roman Empire, it will be only a legalistic or technical point. When, for instance, did the Senate as a body disappear? Or the magistracies preserved by Augustus? The latter must have disappeared bit by bit in both Western and Eastern Empires, I suppose.
As to the Turkish claim, the only counter-argument that I can think of is that the Byzantines did not willingly become second-class citizens in the Turkish Empire, that it was a war of conquest that felled the last Byzantine emperor - much as if the Germans or Parthians had been the conquerors in place of the Turks.
But then, as so often in even the heyday of the Empire - or even earlier, in the later years of the Republic - might made right, in terms of who would rule. The Turks were by 1453 CE a European faction, as you point out. And then the "Latins" had already conquered and divided up the Empire, as foreigners, in 1261 CE.
It's an interesting question.