The Classical Greeks conceived of the world as consisting of 3 more or less equal parts: Europe, Asia, and Libya (Africa). The border between Europe and Asia was, indeed, the Hellespont (the narrow strait between the two parts of modern Turkey, now known as the Dardanelles). While we don't know exactly where this worldview came from, it seems to predate the rise of Greek civilisation.

However, the division was much more geographical than cultural. In the Iliad, there is little difference between the customs of the Greeks and the Trojans; they both seem to speak the same language, share the same values, and worship the same gods. In later times, there were Greeks living on both sides of the divide, with a very substantial number of Greek city-states on the west coast of what is now Turkey. At the same time, there were many peoples living in Europe that the Greeks considered "barbarians" with strange and savage customs, like the Thracians (who lived in what is now Bulgaria, Northern Greece, and European Turkey) and the Skythians (who inhabited what is now Ukraine). While the Greeks thought of Europe as their home continent, they do not seem to have regarded it as theirs.

The politicisation of the border between Europe and Asia came with the rise of the greatest empire of the Classical world - Achaemenid Persia. Several Greek sources acknowledge that the Persian king ruled over "the whole of Asia". Persian royal ideology laid claim to the whole world, but in practice it would have been easy for the Greeks to consider the Persians to be in charge of the entire eastern continent. Between 546 BC (the subjugation of the Greek cities of Asia Minor) and 507 BC (Darius' expedition against Thrace and Skythia), this was literally true; the Persian empire covered all the Greeks knew of Asia, but stopped at the edge of Europe.

However, in 507 BC this balance was shaken when the Persians crossed over into Europe. For decades they controlled large swathes of the Balkans, and before Xerxes' defeat it seemed that all of Greece was going to fall under their sway. After the Greeks' unlikely victory, the anti-Persian alliance immediately began to push back the Persian sphere of influence, turning the tables - and one of their first acts was to sail up to the Hellespont to liberate the Greek cities there. For the next century or so, the Greek cities in Asia were subject to the Athenian and then to the Spartan empire; despite the Persian king's claims, the Greeks who lived in Asia were not under his control.

There was still a certain conceptual grace in referring to the Persian sphere of influence simply as "Asia". When the Persians regained control over the Greek cities of Asia Minor in 386 BC, Greeks began to talk about a panhellenic crusade to liberate "the Greeks of Asia" and to wrest control of some part of Asia from the Persians - usually up to a line running from Sinope on the Black Sea coast to Kilikia or Lykia in the Levant.

Two things are important, though. First, the division was still more geographical than cultural. The argument was that the barbarians happened to rule Asia, but parts of Asia were actually Greek, and they really ought to be autonomous from Persian rule. At most, the Greeks used "Asia" as a shorthand for "the Persian empire"; the border between the two cultures was much harder to define, partly because the Persians left their subjects largely autonomous and did little to change their way of life. Second, the Greeks did not define themselves as "Europe" or "The West". They were the Greeks; they did not represent any greater entity. Indeed, defining themselves as "the West" would be difficult when the far west of the Greek world was threatened by another barbarian enemy - Carthage.

The association between Classical Greek culture and "Western civilisation" is very much a modern idea, fuelled by the desire of Western thinkers to "claim" the Greek past for the West. Needless to say, the idea is completely ahistorical. The Greeks themselves lived in an interconnected world in which ideas, goods and people were constantly moving from east to west and vice versa; they did not regard their culture as "Western" while Persian culture was "Eastern". They did, of course, define themselves as civilised and others as barbarians, but this division no more adhered to the border between continents than the Greeks themselves did. Given the number of philosophical ideas (natural philosophy, mathematics, historiography) and prominent individuals (Pythagoras, Hippokrates, Herodotos) that came from the Asian part of the Greek world, it would be entirely fair to say that there would be no Greek civilisation without constant crossfertilisation between Europe and Asia.

As to your final point - race is a modern concept. The Ancients did not know it. They occasionally remarked on skin colour (they had an odd story about Ethiopians being black because the sun was closer to their lands), but they made no distinction between the peoples west and east of the Aegean. There probably was no difference to remark upon; the peoples surrounding the Mediterranean have intermingled for thousands of years and share many physical features. In fact, I know of only one example of the Persians being mocked for their skin tone, and it is actually because they were too white:

Believing that contempt for the enemy would kindle the fighting spirit, Agesilaos gave instructions to his heralds that the barbarians captured in the raids should be exposed for sale naked. When his soldiers saw them, white because they never stripped, and fat and lazy through constant riding in carriages, they believed that the war would be exactly like fighting with women.

-- Xenophon, Agesilaos 1.28

The mockery is due to cultural differences. While the Greeks of Agesilaos' army prided themselves in their hardiness and exercise, the rich Persians they had captured were used to a life of luxury. While Greeks exercised naked, nudity was a great taboo in Persian society, and wealthy Persians went about fully clothed at all times. As a result, the Greeks who had fought with Agesilaos for a long time were wiry and tanned, while the Persians they had captured appeared white and pudgy.

However, these were just some Persians, from the ruling elite, captured on the fringes of the empire, in a region that was actually Greek, even though it was in Asia. Again, there is no association here between Asia and otherness. It is the barbarian who is strange, not the land he has conquered.