Socrates taught Plato who taught Aristotle. What Aristotle in particular taught and wrote became the basis for Western thought. Presocratic Greek thinkers were the first in recorded Western history to adhere to any kind of critical examination of the world as a basis for their thought, and thus set the stage for Socrates. The Presocratics tried to explain the universe in various ways, but without a means of testing their explanations. Socrates himself simply took this a step further and questioned everything, declaring that he was the wisest man in Athens because he alone was aware that he didn't know everything about anything.

Basically, in Western thought, these guys are where everything started. It wasn't necessarily all superstition and BS before them, but they were the first to really study and try to figure out how things worked based on observing the world.

Eastern thought I'm less familiar with, but I know that Aristotle had less of an effect on the development of science and philosophy in the Far East.

I have somewhat less knowledge about Eastern philosophies (and particularly the history of them) beyond having taken a survey course on Hinduism and having a broad understanding of the relationship between the three major religions.

Edit: Thought I was somewhere else, fixed the self-dumb.

Edit 2, addressing Question 4: As implied above, what modern (well, early-modern) philosophy took from the Greeks wasn't specific conclusions but more a method that evolved over time from what the Greeks did to the scientific method we know and love today. Aristotle in particular tried to explain physical phenomena in terms of cause and effect. To the point where some of his classifications and descriptions of medical conditions are still used to this day - note specifically that the causes (and treatments) of those conditions are not mentioned, because he generally got those wrong. In general, he had the right idea, but didn't know how to test his hypotheses empirically like Newton would figure out later.

To go back a bit further to show just how strange and groundbreaking Aristotle's method was, and why we look specifically to him (and his teachers) as we do, it's worth examining how the Presocratics thought of the world. So these guys came up with the idea of explaining everything (life, the universe and everything) in terms of natural phenomena. No gods, no spirits, no supernatural explanations for things we can observe in nature. So, of course, they came up with ideas like "everything is water, in some form or other" (This was Thales, the first of the Presocratics who lived ~625-550BCE - basically, we can observe water freezing and boiling, and plants use it to grow, and we (or our food) eats the plants, so everything is just a process of transforming water from one form to another). Anaximander came next, rejecting "water" and in its place putting "the boundless", then his pupil Anaximenes said that was dumb and of course everything must be composed of air (because condensation and rarefaction). Separate from these guys were other groups and thinkers, including Pythagoras (yes, [that] (http://en.wikipedia.org/wiki/Pythagorean_theorem)Pythagoras, whose thoughts ranged from humans having fallen from a state of purity (Empedocles, echoed later by Plato's Forms) to the Atomists, who thought that matter could be broken down to some smallest form beyond which it could not be split in half.

Getting back to Aristotle: In his writings, Aristotle claimed/acknowledged* that Thales based his theory on evidence and reasoning about that evidence. He contrasted what Thales said to Hesiod, who had a cosmos origin story that involved the Olympian Muses - basically a creation myth.

To put all of this in a rough time frame, Hesiod was in the century before Thales, and Socrates was about a century after Anaximenes - so in total we're talking 300 years from Thales's birth to Aristotle's death, and no more than 100 before any of them when nobody really considered there might be actual physical explanations for everything. To reiterate, the idea behind all of this is that these philosophers were the first to attempt to describe the world in terms of physical phenomena, without (too much, or overt) claims to supernatural powers. In a very real way, this was the start of a scientific view that we might recognize as such today.

Sources: Mostly A Presocratics Reader, edited by Patricia Curd, also various Aristotle writings I've been reading lately.

* We don't have any surviving complete writings by any of the Presocratics, and pretty much what we know of their thought is from fragments, quotations and discussion by later writers, including Aristotle.