It actually has been studied, and DNA evidence has pretty solidly demolished Arthur Koestler's thesis that most Ashkenazi Jews are descended from the Khazars. Ashkenazi Jews have numerous genetic markers showing them to be more closely related to middle eastern Sephardi Jews than to non-Jewish Europeans.
The Khazar Kingdom nominally embraced Judaism as a political move-- to avoid being swallowed up by its larger and more powerful neighbors, Christian Russia and Moslem Turkey. But very few of the people actually embraced Judaism in more than name. And even the nominal conversion lasted only one or two generations.
By this I mean that from the surviving records we have, they had a fairly sophisticated level of civilization. We know that they were able to keep the Byzantines, Cossacks, and Golden Horde back enough to get breathing room, and could project enough power to have favorable treaties at times. But we know very little about them.