a distinctly Turkish entity,
Perso-Turkic is IMO a more accurate description - the Ottomans identified more closely with the settled Persianate empires of the East (although they did indeed base their empire in Rumelia), not with the illiterate peasantry.
Ottomans demanded Austria to humiliate itself and only call the Sultan emperor, not using the imperial title for itself when dealing with ottomans until peace of Karlovitz.
Charles V and Ferdinand I certainly addressed the Ottomans as the Caesar of Rome.
This is a misunderstanding of the affair. To the Ottomans the Sultan was superior to all monarchs; their monarchy was theoretically universal (yes, this was a Greek influence, though likely not directly). To recognize the Holy Roman Emperor as an Emperor - a title which also connoted universal monarchy - was to make him equal to the Sultan, when to the Porte the Ottoman equivalent of foreign monarchs was the Grand Vizier.
So claiming no one except Mehmed cared is not correct.
The Trapuzentines, the Austrians, Spanish, French, Russians (on an on and off basis), Iranians and Chinese recognized the ottomans as the legal successor of the Byzantines. Who were the successors of Rome...
It's not that Mehmed was the only one who cared; he was the only who seriously had pretensions of restoring the Roman (=Byzantine) Empire and was the only one who fashioned himself as a direct successor to the Roman Emperors, from Augustus to Constantine XI. The others merely treated it as a principally geographic title; the Ottomans were part of the Roman geographic sphere to the rest of the Islamic world (and to China), and their heartland was indeed
Rum (=Rumelia), while the birthplace of the dynasty was Anatolia (also considered geographically part of Rum). To tell a Sultan that he was not
Rum would have dumbfounded him; he would not have had the ability to see Rome through a faux-Latinate veneer.
Ottomans took from Byzantines the food, music, court and administration, architecture and a decent chunk of philosophy.
I don't see how being influenced makes the Ottomans 'Romans' -- they were also influenced by the Arabs and the Persians, after all.
Ottomans were the same mixture of population that Byzantines were with a bit of Turkic added in.
This does not make them a continuation of the Roman Empire -- all it means is that the Ottomans considered themselves to be ruling over
Roman lands (without actually identifying as 'Roman' themselves).
... big antagonist for centuries of Christian Europe.
Not really, no: they were principally the 'antagonists' of Austria and her allies, not all of Europe (France, for instance, had a strong alliance with the Ottomans well into the 18th century).