Is there a relationship between these folks and the start of civilization? Mehrdad Izady suggests that they started civilization via establishing the agricultural production. How can this be possible, the Kurdish area is within a severely mountainous geography? And why they couldn't have any important impact in history like Greeks or Persians?
It is certainly true that many of the foundations for later civilisations, like the development of agriculture, the development of writing, and even an invention of the wheel, can be traced to The Fertile Crescent.
It is likely that many of the ancestors of modern Kurds came from the the Fertile Crescent. It is quite possible that "The land of Karda", mentioned on a Sumerian clay-tablet from the 3rd millennium BC is the earliest reference to a "Land of the Kurds". If so, the Kurds might have originated from an area to the south of Lake Van. However, we must remember that the development of agriculture was perhaps some 5000 years before the development of writing. Any suggested link between a possible Sumerian mention of a "Land of the Kurds" and the people that invented agriculture 5000 years before is tenuous at best.
So we can reasonably say that the ancestors of modern Kurds came from an area within the Fertile Crescent, although - based on recent DNA analysis - we can probably say the same for many of the ancestors of peoples from Europe, North Africa and a large part of Asia. We can also say that many of the foundations of civilisation were developed in the Fertile Crescent. But it would be going too far to claim that any one of the groups of people in that region "started civilisation".
That seems highly unlikely.
The invention of agriculture in the Fertile Crescent was roughly 10,000 years ago, principally in the Tigris-Euphrates river valley system. The people living there when the historical record opened spoke a language unrelated to any modern language.
Kurdish is an Indo-European language, of the Western Iranian branch. About 5,500 years ago all of these languages would have been a single language spoken by a single ancestral people (who we call Proto-Indo-Europeans, or PIE). Our best guess is that they were living north of the Black and Caspian seas, on the Steppe. It is quite likely these are the people that domesticated the Horse.
The Iranian branch split off when a group of PIEs moved into Persia, likely from west of the Caspian. This happened about 4000 years ago.
So no, there aren't really any reasonable (standard) interpretations of history that would credit Kurds with Agriculture in general. Its possible their ancestors could be credited with domesticating the Horse, but no more so than the ancestors of the English and Indians can (same ancestors).