Although the question is too broad in scope to be answered on this site, I shall attempt to address some of the points raised in the question. To summarize, as is often the case, the importance of Greeks is most attributable to the reason that often dominates most of the history -- coincidence.
When, We open the history books we find, that: Greeks have greatly succeeded in a lot of fields and sciences (such as: philosophy, medicine, maths). In addition, to wars
In the aspect of sciences and philosophy, Greek civilization was not particularly "ahead by miles" compared to its contemporary civilizations. At least on top of my mind, Indian philosophers, mathematicians and medicinal experts were on par with their Greek counterparts. But as it happens, the Western Civilization inherits its culture from Greece.
As for the wars, if you trace the trajectory to understand the current situation in Europe (or, let's say, why WW2 happened) -- the oldest well-documented history is that from Greece. That is not to say that other civilizations did not have a significant history or they did not document it. It's just that Greek history is most relevant for understanding European history.
The most great commander was Greek [i.e. Alexander The Great] even "Julius Caesar" and "Solomon The Magnificent of Ottoman" were jealous of him for achieving greater goals than theirs when he was just 25-years-old
Alexander the Great was, again, one of those accidents that changes the course of history. It just so happens that he was Greek (technically, a Macedonian). And he did have an enormous impact in the whole Mediterranean, Mesopotamian, and Persian regions, in process, Hellenizing them.
The most (brave, valour, fierce) soldiers were "Spartans" they were so much fierce and strong that they once killed 22K Persian soldiers with only 300 Spartans
Noting on side that "only" 300 brave Spartan soldiers killing 22,000 Persian soldiers is simply not true, it is indeed true that Spartan had a radically different and harsh way of life than any contemporary state. However, it is not clear how this unique way of life used to measure "success" of the Greeks.
Even their letters [i.e. Greek alphabets] are used in lots of sciences (such as: physics & maths)
It is important to highlight (a rather obvious) fact that even though we use Greek alphabet in the sciences, most of the European languages do not use it in their scripts -- those are Latin alphabet or some variations thereof. So the most ubiquitous alphabet is Latin, and not Greek. And since the (Western) European scientists were using Latin alphabet (and until recently, even the Latin language) to "write" the sciences, they turned to the second closest alphabet that they were familiar with -- Greek -- for symbols.
So to summarize, Greeks have a perceived success only because they have an unparalleled cultural position in the Western Civilization, and Western Civilization happens to be the most dominant one at the present, in terms of cultural as well as science-technological advances. Furthermore, this is not a direct influence, but rather an indirect one coming via Romans who were Hellenized when they came in contact with the Greeks in the 2nd Century B.C.
However, this is not to be mistaken with Greeks being a continuous historical anomaly for the 3-4 centuries that the Greek civilization lasted. Contemporary civilizations like China and India were comparable to the Greeks, but it just so happens that those civilizations have not influenced the Western Civilizations as much as the Greeks.