(to simplify)

Don't.

middle eastern genes

Do genes have a passport?

How do you tell if the genes are «Khazarian», «Aryan», «Roman», «German», «Slavic» or anything else? Why would you be particularly puzzled by the genes of the Jews? Christians aren't native to this region either. Back in the Roman era you had all kinds of heathens in Germany, including Mithraists, including the cult of Isis. You had soldiers from the entire empire. You had merchants from the entire empire. And yet you're only puzzled by the Jews.

To Zionism it is irrelevant where your geneticks are from. Herzl claimed «The Jewish state» was made necessary not by some historical motherland but by the Christian nations, all not extremely educated on the topic, deciding which of their minorities belong to Europe and which don't.

The Khazarian extraction was first proposed by Arthur Koestler in the «Thirteenth Tribe». It is well known the French Jews were assimilated by German Jews. It is reasonable enough to presume they have assimilated the Khazar Jews as well. And the Germans, and the Slavs, and the Huns, and the Asiaticks, and the Africans and everything that came into their path.

Because Judaism does not forbid conversion. Missionary work is banned. But if you want to become a Jew, as it happened to the Khazar elites, nobody holds you from becoming one. And the first Jewish Khan of Khazaria has, by halakhic law, been just as "Jewish" as Moses himself.