Greetings, I only have a couple of "simple" questions: When did the Greeks actually start calling themselves Greeks? Wouldn’t that require the establishment of a nation or country called Greece? When did these “city-states” collectively call themselves Greece? Did they call themselves Greeks before they were one nation? --- clarification – as far as I am concerned “Greek & Greece” are the English translation of the words Ελληνες & Ελλαδα. I do not consider them two different words as most Greeks do (and I am Greek). While I was living in Greece I was told by some learned Greeks that it was the Romans who first called them Greeks, when they conquered and established the Roman Empire. There seems to be a great debate over the origins of the word “Greek”, the derivative of the Roman word for Greek, etc. – I believe that even when the Romans and all other nations began calling them Greeks, they did not call themselves Greeks. I am under the impression that Greece became a country after they won their freedom from 400 years of Turkish rule in 1821. Therefore that would also mean that this is when they actually called themselves one people, Greeks or Hellenes if you prefer.
So far as I know the first testimony that Greeks called themselves by a single name, that of "Hellenes", is available already in Thucydides (1.3.3-4), and refers as back as to the times of Homer -- with Homer himself referring to an even more distant past. You can read more on the origins of Greeks, including the name, in my book on The Ancient Greeks (check especially pages 137-139).
The Romans used the specific word 'Greeks' instead of Ἕλληνες, because they first knew them from immigrants in Italy coming from a Greek city called Graia, not after they conquered Greece. In any case, this refers to the specific epithet, it doesn't mean that at this time Greeks / Hellenes did not recognise each other as a single people of a common origin under a single name, that of Ἕλληνες.
Regarding your other question, "wouldn’t that require the establishment of a nation or country called Greece?" No, it wouldn't necessarily. Think even about today, with Greeks living all over the world, from the US to Australia. They are all Greeks, they call themselves Greeks, but they do not belong to the same country. The same way in ancient Greece, the various city states understood themselves as Ἕλληνες, although they belonged to different states / countries.
Of course, you should force yourself to improve your Greek! :) An English version is not something insignificant, considering how many people speak English, but a translation needs a lot of time that I don't have, at least not now.