I have some time to kill today, so let me clarify my views on some issues:
My reasoning behind suggesting razing many of the Mongolian cities involves trying to maximize the potential of the captured land. This is important IMO if the rest of the game plays out peacefully since the cities you end up with after the Mongolian war will have to see you through to victory. What may be important to decide at this point is whether cultural or spaceship is the intended victory type as this will effect the decisions on whether to raze or not. My earlier recommendations were made based on pursuing a space race win.
Regarding Turfan, yes it is a good city site. But its location also means that a number of tiles either won't be worked or will be worked by a city that is squeezed in with major overlap and limited potential. By razing Turfan it allows two good cities to be built in that space. One built 2 tiles SW will be a strong production city, which will be helpful in producing military units if other wars are to be fought. But more importantly it will also be very useful when it comes to building spaceship parts. The other city built 1 NE of Turfan will be a decent commerce city (although not as good as Turfan). Overall this is a better option when shooting for the stars. However if the decision is to go cultural then Turfan might be a good choice for one of the three culture cities because of its high commerce potential.
Old Sarai will make a good production city with six hills and a banana plantation to feed the miners. But moving it one tile E keeps all of those tiles and adds a seventh hill. If OS had a high population at this point I would recommend keeping it, but it will go to size 2 when captured, and its unlikely there will be any buildings so really there is minimal cost in relocating it to make it a better city. Again, for space race having high production cities is important.
Ning-hsia was just a badly placed city IMO that should be moved. Exactly where depends on whether or not the clams can be used by another city that claims the copper. If Mehmed hasn't founded a city to claim the copper by the time Ning-hsia is attacked I think it would be a good idea to have a settler ready to found a copper city immediately (in fact I'd wait to attack Ning-hsia until the settler is there). Although you have iron, remember that copper speeds up some builds
including space ship parts.
One major point in favor of razing is that it will help financially because some of the city sites do not have to be refounded immediately. The two Turfan replacements and the Ning-hsia replacement could really be left for a while until finances have stabilized a bit. The Old Sarai replacement might also be able to wait if Kumbi Saleh and Hittite both get border expansions soon (and they should).
As to the debate over CoL vs. Currency, I think if you ignore trade opportunities for the moment Currency is clearly the more immediately beneficial tech. You would get income from extra trade routes without needing to spend any hammers (which is good since we want the hammers to go for more military at this point). When you can afford to put some hammers into a civilian build a market in Timbuktu will give a sizable return since that city is generating most of the empire's commerce. And Currency is a pre-requisite for CoL so that tech would become cheaper. I don't really see any immediate financial benefit from CoL since cities would have to build courthouses and none of the outlying cities really are strong production cities. Also I don't know that city maintenance will be all that bad, especially if you raze some of the cities and wait a bit to rebuild. CoL does make caste system available, but I don't think you're going to be in a position to take advantage of it any time soon.
So the only reason I can see to research CoL first is if you're confident it will offer better tech trades. One question I would ask is if you research CoL first and find you can't trade it for Currency at that point would you research Currency yourself or research other techs and wait for a trade to become available? It would seem to be a waste to research a tech and follow it by researching one of its pre-req techs.
Finally we have the cultural war over the pigs.

My view is that you've lost this round. Seville already has a monument, library, Buddhist temple and Buddhist monastery plus getting culture from having the state religion present. You're not going to be able to flip that tile any time soon simply by constructing culture buildings. In the not very distant future there will be some other options available that will get the job done. Music will allow building culture and will also make cathedrals available. And there's the possibility of a great artist or building a wonder in KS. That's why right now I would like to see KS's hammers go to support the war effort in some way.