If possible, I'm most interested in what kind of syncretic religion/culture emerged from Tengriism/Judaism touching and what the lasting effects were. That said, I'll happily consume any favorite facts or illuminating chunks of information about the Khazars in general.
This is a big and interesting topic that hasn't necessarily been explored on /r/AskHistorians in the detail that it deserves. While you are waiting for answers you might be interested in reading some previous threads on similar topics for the background info:
Who were the Khazars and where do they fit into European History? Why did they convert to Judaism? - 11 comments, over 1 year old.
The topmost comment provides a good overview of the Khazars in general and also discusses the possible reasons for their conversions to Judaism.
What happened to Khazarian nobility after the fall of the empire? Was David Al-Roy a Khazar? - 7 comments, over 5 months old.
/u/gingerkid1234 gives a strong, sourced answer on the "Khazarian hypothesis" which deals with the possible connections between the Khazars and the Ashkenzi Jews.
Origins of Eastern Europe's Jews? - 6 comments, over 6 months old.
/u/gingerkid1234 once more talks on the topic, this time giving only a brief summary of the Khazars in relation to Judaism and talking more about the history of Judaism in Eastern Europe.
The King of Khazars is said to have seen himself as a "defender of Jews everywhere". In the Early Middle Ages, would the Jews of Europe have known about Khazaria? Were there any recorded instances of European Jews migrating to the Khazar state, for a better life? - 8 comments, over 9 months old.
There were a fair number of top level answers in this one, but there wasn't as much discussion or evaluation as other.