I think all of your questions are not referring to the same topic (i.e. art, politics, culture), which will make my answer a bit difficult.
- Title question: did Egyptian/Mesopotamian society have influence on Greece? I would say greatly. To begin, art in Ancient Greece went through phases of development: neolithic, Cycladic, Minoan, Mycenaean, Helladic, Geometric, Proto-Archaic or Orientalizing, Archaic, Classical, Hellenistic. Now, Orientalizing means exactly that: in a period of a hundred years, between 700 and approximately 610 BC, Greek merchants were very active with the Levant, to a lesser extent, Mesopotamia and Egypt, where they brought back metal goods from the Near East, back home, where it was imitated by artists and artisans in the production, painting and something very important, writing, which the Greek (and Latin, for that matter) alphabet was adapted for the purposes of the Greeks. Along with this, myth and its depictions, such as the griffin, the lotuses, various geometric shapes found on pottery, evoking the metal works brought back to Greece had an enormous influence on Greek culture.
On the other hand, Greek politics remained static during this period.
Yes, Greek culture was enormously influenced by the near east. After the battle of Marathon, the Persian Wars, there was a very deep psychological break between Greece and the East. I believe it to be not have been repaired in the preceding close to 3000 years.
Let me know of any other questions. I am not entirely an expert on this period of Ancient Greek/mediterranean history.