It doesn't look like Khazaria really has to "reclaim" their empire; they just have to recover further from the Kabar uprising, and, most importantly, reestablish contact with the Byzantines.
Seeing as the Kabar rebellion was a failure in OTL (they were driven out by the Khazars), I don't see how it could have been resolved differently. A better way to keep the Khazars strong would be to avoid the Kabar rebellion and keep the Magyars living in Atelkuzu (in OTL Ukraine and Moldova).
Same here, except for a few other sources.
Anyways, I wonder if I could have the Jewish rabbis that come to Khazaria tone down the nomadic lifestyle of the Khazars, making them focus more on Itil then before?
They were pretty urban in OTL anyway. The Khazars would spend most of the years living in their various cities (Atil, Khazaran, etc) and would occasionally go out into the countryside to work in agriculture, wine-making, herding, and other things you'd expect of a nomadic tribe. But the focus was generally on the urban life.
I tried to get them to survive in Raptor of Spain but.... it's not happening. They're dying in the late 11th Century. It's the Highway issue, they have to keep fighting off waves of eastern steppe peoples but are also becoming more urbanized as a whole and thus weaker against them.
So essentially I agree with Valdemar II. The Khazars essentially ran out of steppe and so had to fight every single wave of eastern invaders and while they did well for a while eventually events are going to turn against them all at once (unless they remain lucky for 500 years) and they will be overcome.
Honestly, this does not seem like a huge issue. There was plenty of land nominally controlled by the Khazars, who really just needed cities anyway. The encroaching Patzinaks / Pechenegs did not trouble the Khazars until the khaganate began to decline in the 10th and 11th centuries. With a strong central authority, perhaps the Khazar ruler could dictate to the Patzinak (and eventually Cuman / Kipchak) leaders to settle somewhere in the plentiful plains of the Ukraine.
Russia are very hard to avoid, it's well placed for unification and expansion, of course it also had a lot of luck in the Poles more or less self destructing, but expansion into the Don and Volga watershed are almost unavoidable. Of course lots of people are still alive in the Caucasus even with the Russian conquest, and we could easy see Khazars survive to modern day in the area as the Ossetians, Armenian and Georgian has done.
One idea I had was for the Khazars to subvert the growth of independent Russian states by keeping Kiev, preventing the Varangians from taking the fort. This would force the early Rus' to trade down the Volga, leaving them dependent on the Khazars. They would have to pay tolls and whatnot, and perhaps become more culturally aligned with the Khazars. This idea vaguely ended with the early Rus' states becoming part of Khazaria, which resulted in an empire linguistically a mix of Khazar Turkic, Hebrew, and Slavic.
