1) They would be Athenian subjects or they'd be torn to shreds and their cities razed to the ground for objecting.
2) The issue is that people define Rome in manners that suit their in-built ideological views more than historical accuracy. Historical accuracy would note the Ottoman emirate had more connection in a direct sense with the Roman Empire of the 15th Century than say, Russia or the HRE did.
1) sure, but that has nothing to do with what we are discussing. The point is that the roman colonies were, by their own nature, very different from the greek ones. Roman colonies were mostly a mean to control a territory through a fortified center. The colonies were splinters of Rome scattered among italy, unlike the greek ones, which were increasingly autonomous entities. Rome wasnt just a city state whose population fled and created colonies, it was a growing state which actually controlled with its own population and increasingly "roman" allies a region far bigger than its walls and the sorroundings territories.
2) you say so. But as i pointed, there are few if any similarities between rome, even the dominatus (dominate?

) , and an empire which whose language, burocracy, religion, law, military sistem were completely different and which (ulike the ERE) couldnt even claim a legal or moral succession from constantine which just happened to claim to be roman. id also point that unlike christianity which was a religion which, even if radically different and often opposite to the civic roman beliefs, developed and spread inside the empire, managing to affirm itself thanks to the empire, ottoman islam was a cult which was been an enemy of the empire since its birth, and that was imposed in constantinople by a foreigner invader army. I wouldnt call it a little difference.
Neither the russians or the HRE were romans, for sure, btw; though the western and eastern goths and the frankish kingdoms which replaced rome in the west were extremely romanized, so that it could be easier to call theodoric italy roman than the ottomans.