The Jews of Europe may well have known about Khazaria. A major Spanish religious text was named Kuzari in the 12th century. It's a book outlining Jewish beliefs, framed as an explanation for the Khazars. The Khazar correspondance and Schecter letter also indicate a degree of contact between Khazars and Jews elsewhere.
As for migration, I'm not aware of any mentions of it, but records of Jews from that part of Europe are quite spotty. It's a possibility, but there aren't any recorded instances to my knowledge.
To be clear, the system of power in Khazar state was dualism. There was a king, and there was an "underking" with broad prerogatives. Historians still haven't confirmed were Bulanids (Jewish convert Khazar dynasty) kings or underkings of Khazars. This isn't offtopic - the problem is, that the other dynasty (analogously kings or underkings) was not Jewish at all, probably Tengri or even Orthodox. The question arises, was Khazaria really the heaven on earth for Jews?
Like /r/ginerkid1234 told you, the most important source is the Iberian Khazar correspondence. We also know that many rabbis in the Holy Roman Empire (Germany) knew about Khazaria, and you should check out history behind the https://en.wikipedia.org/wiki/Radhanites , the medieval Europe-Asia Jewish commerce network which fell after the collapse of the Khazar and Tang China states. One of the traveller rabbi: https://en.wikipedia.org/wiki/Yehuda_HaKohen_ben_Meir
I doubt Sephardi Jews of Al Andalus, in the time of the Khazar khanate (7th to early 11th century), would be willing to immigrate en masse and left their blooming commerce and society in Spain to fricking Eurasian Steppe which was the Khazar realm. What we have was Jewish merchants that do traveled as traders between Islamic Spain and Arabia, Byzantine Constantinople and Khazar khanate.
Sephardi Jewish society and Cordoba caliphate in general was at peak of their power and prosperity, so it was not logical to Jews to migrate to the sparsely populated Khazar kingdom, which consisted mostly of steppe between Black sea and Caspian sea.
There was large-scale immigration of Jews from both the Byzantine Empire and Persia into what became Khazaria in the seventh century (fleeing persecution by the Heracleid emperors and the Mazdakites), so much so that the city of Tamatarkha (near modern-day Tmutarakan) became known as "Samkarsh of the Jews" in Arabic. The Schechter Letter (written in the mid-tenth century by a Khazar official) mentions immigration of Jews from Armenia at some point in the unspecified past. But this would have been before the Khazar conversion to Judaism. I'm not aware of any large-scale immigration of Jews after the conversion (i.e., in the ninth and tenth centuries) but that doesn't mean there wasn't some migration. Interestingly, the letter of Hisdai ibn Shaprut to the Khazar ruler Joseph says that Hisdai would leave all of his possessions and power in al-Andalus for the chance to stand in Joseph's presence: "If indeed I could learn that this was the case, then, despising all my glory, abadndoning my high estate, leaving my family, I would go over mountains and hills, through seas and lands, till I should arrive at the place where my Lord the King resides, that I might see not only his glory and magnificence, and that of his servants and ministers, but also the tranquillity of the Israelites. On beholding this my eyes would brighten, my reins would exult, my lips would pour forth praises to God, who has not withdrawn his favour from his afflicted ones." Joseph's reply invites him to do just that, but there's no evidence that Hisdai actually followed through (probably for the reasons reddripper states below). As far as the Khazar rulers seeing themselves as "defender[s] of Jews everywhere" - there are at least three key pieces of evidence for this claim: