The Khazar rulers were Turkic peoples who ruled what is now southern Russia and the Ukraine for centuries around the birth of Islam. They were turkic, not caucasian. They ruled over Slavic, Germanic, Irani (Alan), Ugric peoples, but there is no evidence that these ruled-over peoples ever converted to Judaism
The elites converted to Judaism -- it is conjecture to say they did it for political or economic reasons -- they could have stayed with shamanism or Tengriism
Also note that European Jews share the same genetic materials as Jews from the middle east, india, etc
Well, you’re wrong on most of what you said. But you repeat the common narrative very well, so I’m sure you got a gold star that day.
Read up on it. Start with Koestler’s The Thirteenth Tribe.