As far as what to do next... My vote would be to continue the war against Khan. There are two primary goals in this effort. Eliminate his source of horses and take Beshbalik. Finding out where is other city is, taking Old Sarai, and taking his capital would be nice, but might require more resources than can be mustered in the timeframe we're looking at.
Severing his supply of horses could be more difficult than you'd think, as Beshbalik is probably stocked full of Keshiks (judging from his still-high military power and the early screenshot). You'll likely need an axe or sword to kill whoever's guarding the horses and then a couple of spears to protect them from the oncoming assault. If the spears die defending the pillager, it's no big deal. They will have, most likely, taken many keshiks with them.
In order to take Beshbalik, you'll most likely need a fair number of catapults. I'd bring 5-6 cats, 2-3 spears, and 3-4 axes/swords. That'll require basically your entire current force plus a couple of cats (and maybe a spear). The cats are probably going to need to suicide to take the city, but, by the 3rd or fourth one, they should no longer be suicide missions. At this point, you may wish you had left the cats un-promoted so that you could give them city raider promotions when needed. This, in general, is one thing you need to work on. Leave troops unpromoted until you know exactly what promotions they're going to need. This allows them to a) better react to whatever situation they're in and b) quickly recover from collateral damage.
I'm not confident that the war can easily be continued onto French soil. His power graph is quite high, which means it's all sitting in his cities. It'll take considerably more catapults (and the troops to protect them) in order to wage a successfull French invasion. On the plus side, he doesn't seem to have any desire to launch an offensive of his own, so there's little risk in attempting to take a city or two. On the whole, though, I'd do a little bit of scouting in his territory before deciding if an invasion is called for.
On the domestic front... Keeping the spices near Turfan is important. Get a spearman on there ASAP. It's going to be a prime target for Keshik raiding parties. Unless you're planning on an invaion of France, I'd lay off the whip for a while, at least until the unhappiness settles back down. You need to start mixing in some domestic buildings along with your military soon, especially in Timbuktu. Decide which cities are going to be focusing on commerce through the rest of the game and start getting a library or something in place there (mixed in with some whipped catapults). Your cities near the front should probably lean more toward military production in the near future. Also, you might consider spending a few turns chopping an additional couple of workers. You're going to need them soon.
This game isn't going to get easy anytime soon, and it's definitely not a foregone conclusion one way or the other at this point. I don't think you've captured enough land yet to stop the warmongering. Taking the entire continent shouldn't be neccessary, but you should at least eliminate one opponent, and cripple the other. Doing that sooner, if possible, is obviously better.