1. The capital looks fine. It can be good to create a canal with the City Center, although its not that big of a deal, especially since you can build Canals now. Remember that your City Center is always at least 2f/1p, so if you found your city on a tile that has less than that, you are actually improving the tile. The opposite is true if you found a city on a really good tile of course. Generally that means waste tiles (dester/tundra) and flat tiles (grassland/plains) are good for founding cities on. Plains Hills are the best because they already give 2p so you end up with a 2f/2p which is superior. Also founding on a luxury and trying to get at least 1 2f/2p tile in your first ring is best practices. You've got a good supply of hills. Food might get scarce (until you harvest the stone).
Queshqer - I'm of the opinion that founding a city or placing a District directly adjacent to a volcano is almost always incorrect. It is almost always better to leave those tiles open so that 1) you don't have to deal with much destruction and 2) you gain the benefit of the extra tile yields after the eruption. It would be better to found it in the desert - ocean access isn't a big deal (you already have it) and again, remember, waste tiles are good in a pinch because your City Center is 2f/1p no matter what. Waste tiles are also good for having empty space you don't mind putting Districts on. Founding cities in Desert/Tundra is by no means optimal, but it doesn't suck as much as it might seem at first! I think the Desert Hill two tiles east would have been a better place.
Ulaanbaatar - Okay so the Paititi Natural Wonder. Its true the Culture/Gold bonus stays with your City Center, but you've made it harder to get the District adjacency bonus. Ghenghis Khan is actually pretty good at Religious Victory (thanks to Ortoo and the visibilty combat bonus) so consider plopping a Holy Site down next to it instead of you City Center. Honestly, though, the actual tile yields are so nice I think they almost balance out the Theater/Commercial bonus. I think there's an argument for putting the city where you did, except if you founded one tile to the northwest you'd get another canal and you would still get all the benefits of the Natural Wonder. I also feel like you want to give Natural Wonders (and Mountains) a one-tile berth as a general rule so it gives you freedom to place Districts there later or to benefit from their tile yield bonuses.
2-3. I have no idea what your build order is. You got Political Philosophy pretty early but I think that is mostly because you got Paititi. I feel like you want an early military more than you want that Warlord's Throne early on. I always ask: how am I going to benefit from the thing I'm building *right after I build it*? If I'm not going to benefit, then maybe I'm building it too early. Genghis still needs infantry, at least to start - if you didn't get as lucky and you had a couple of aggressive Civs or a Barb camp spawn on top of you, you'd probably be dead. I'd like to see Bronze Working and some Encampments up and ready for Ordus. Militaristic Civs really benefit from tech beelining. For Ghenghis, that means beeling Bronze Working and Horseback Riding, and forget about everything else until that's done. I mean, you have Pottery-->Writing and no Campuses up, so that's basically wasted Science right now.
For early Militaristic Civs, you want to get Political Philosophy (for Oligarchy, but remember mounted units don't benefit from the +5 bonus), and you want Military Tradition (for flanking) and then it kind of doesn't matter what you do for awhile. The Colosseum can be a particularly helpful Wonder for aggressive Civs, and Divine Right has a ton of Military Policy slots (which makes it good for building a navy and/or upgrading troops). Veterancy (Military Training) is really good because it rushes Harbors (boosting your gold income).
Okay, to sum up: I'd say you are putting the cart before the (literal) Horse here. You already have Political Philosophy, which you couldn't help because you were handed a massive amount of Culture, but you built your Plaza already but you don't have any military which means 1) you aren't going to benefit from WT for a long time and 2) if you hadn't lucked out in the neighbor department, you might be on your back foot and fending off AI invasions right about now. Build military first, then build the things that buff them (PP and your Plaza). You'd rather have an un-buffed military than having buffs doing nothing if the AI catches you with your pants down. Tighten up your tech research when playing a Civ like Mongolia with an aggressive early UU. Mongolia specific tip which you haven't gotten to yet: Mongolia's UU is not the Keshig, its the Horseman. Keshig are nice, but Horsemen should be the main focus of your army. Maybe even just one for hauling your GG around.
