You're correct that Ancient Greeks would have identified with their respective city-states and not with Greece as a whole, but they did nonetheless recognize a sort of bond or special relationship with each other. Homer refers to the Greeks collectively as "Achaeans", "Argives", or "Danaans", but it isn't until after the death of Alexander the Great that there is a unified Greek culture to speak of. What largely unified them before Philip II of Macedon (Alexander's father) subjugated Greece was the pantheon they shared. Sometime after Homer, the Greeks took to calling themselves "Hellenes" after Hellen, mythical son of Deucalion and Pyrrha. "Hellas", in turn, was their word for Greece, and this collective term is the reason the post-Alexander period is called the Hellenistic Period. Thucydides writes about this in Bk. I of his history of the Peloponnesian War.

What united the Greeks during the Persian Wars was a threat to every Greek city-state, not a sense of duty or obligation to each other as neighbors or fellow Greeks. The Greeks' reluctance (or perhaps inability) to conquer each other is something that mystifies me, to be honest.

They did indeed speak and write in the same language and there were also various dialects. The three main ones historians recognize today are Aeolic, Doric and Ionian Greek. If you were to learn Ancient Greek, you'd begin by studying Attic Greek--the sub-dialect of Ionian specific to Athens. This is because most of the literature that survives (and seems to have been produced) was written in Attic. This dialect, by virtue of Athens' special position in the Greek world, came to be the dominant one both spoken and written. In fact, Koiné ('Common') Greek developed out of Attic during the aforementioned Hellenistic Period to become the lingua franca of the Mediterranean and is also the language of the Christian New Testament!

This website (you'll have to scroll past the tables at the top of the page) has a helpful color-coded map of the Greek dialects. The author of this webpage also provides an example that I'll quote:

The word for "day" in Attic is hêméra (with the circumflex simply to indicate the long vowel). This contrasts with haméra in Doric and êmérê in the Ionic of Ionia.

And then, of course, there is Homeric (or Epic) Greek, in which (you guessed it!) Homer's epics as well as the Homeric hymns were composed. To address the very last part of your question, written dialect was often more a matter of literary convention than where one came from. As Attic became the de facto literary language, writers often adopted the peculiarities of Attic Greek ('Atticisms') or even straight-up invented them.

Frankly, that's pretty much exhausted what I can say about that, but please do let me know if you have any further questions. I could at least maybe point you in the right direction!

Cheers.