If it's 1205 already, you shouldn't go to Khazar. Get Kiew and fortify. It won't be long before the Horde appears...
Don't worry, I'm not planning to keep Khazar (or Kiev for that matter). The Mongols can play with the Turks and the rebels before I'll bother dealing with them. Hopefully by the time they come, my troops will be nicely built up.

Beating the GH in Khazar is quite doable if you have the right troops to do so but beating them when they attack from Khazar to Kiev is way easier. Honestly if you have a few battles with the GH each turn and have on top of that to face a Papal resurrection, you are in it for a treat if you love very long battles (most of the time, you can never win against the GH if you auto-calculate).
In my last campaign with the Italians, I took down the horde when they showed up only in Khazar, and I took them down in one battle. Toughie, with some 5k slain, losing some 2700 myself in wave after wave. You can take them down with lower tech troops, but it's a challenge.

Speaking of the pope, just took him down a few years back, and haven't seen him yet again. He seems to favor returning in Rome, time and time again. I kind of like that province now, so I have more than a few troops there. He did also show up a few times in Tuscany and Naples, but not recently.

Another reason I don't have time to be too concerned with eastern expansion plans is that I have other problems. Not following my own advice, I knocked off the Hungarians accidentally (I usually like to leave factions in one unimportant territory, as I know where they are, and usually they can't expand easily with only one low income producing province)). The Hungarians have now returned in Carpathia in numbers, though a small battle with rebels might dwindle them a bit. I now am in the process of beefing up some areas near by depending on which way they expand. I have already sent troops to Bulgaria, Serbia and Croatia, but I will need more troops possibly for Austria and Bohemia as well. Of course the Polish stand in the Hungarians' path to Austria and Bohemia, so I'll probably be okay there for a bit.

I also have been busy employing spies in ravaged provinces I didn't want to bother to keep (though the spare cash came in handy). I will also have some free troops coming from Trebizond as I don't plan to hold that province either. I already had a resurgence of Byz in Khazar, but I have rebels in between them and me before they can contact Costantinople. My only direct contact with the Byz is in Volhynia, but I have lots of troops in Lithuania to keep the peace.

I found I actually had more in the bank than I had reported, some 70k (now in 1209 down to 50k)! I spent some of that on pushing some troops out, mostly arbs and xbows, and adding some trading and farming as well as building some higher tech troop buildings (halbs soon will be online). Still have some trade income to play with, and finally adding caravels as well to replenish a few ships I lost earlier. I have the only navy now, I destroyed most of the other factions' ability to make ships, and the ones that could don't really have the cash or provinces to expand a navy quickly.

Now, my only other worry with surprise appearances is with the French, the Argonese, and the Italians, though the French have been gone a long time. The Argonese disappeared one turn after they were rumored to be attacking me a few years later. I was wondering how they would do that since their king was being held for ransom by rebels that very turn. The Italians I hope will return again to Corsica, as I have it 'spy laden' and have the neighboring Sardinia well fortified and can easily bring in extra troops by boat on my well controlled open seas.

Never a dull moment in this game, that's for sure. Always some surprise to deal with.