Sorry, I wasn't sure whether I ought to have posted that much info here or not (as this is the 'Quick Answers' thread), but I also felt that this question was not worth starting a whole new thread in the strategy section, so I came here.
The strategy and tactics forum is an excellent place to post savegames and ask for advice. There are lots of good players there who don't mind looking at a game and giving advice. But a savegame is really needed to give decent advice. Looking for a few minutes at your savegame gave me a lot more information than what you could have told me in an epic length post. A screenshot can also be useful because it is more easily accessible than a savegame, but a savegame will give far more information.
This thread is mainly used for quick tips so it's not really ideal for this type of game-analysis, but it's not a real problem if it doesn't happen too often.
Here, I'll post the savegame:
Some Notes, First:
New Sarai is quite a problem; It's very isolated, and the Mongolian culture around it is too strong for me to reconnect it with my land without a great artist, probably.
Yes, but it's just one city. It's not really important in the big picture. There are a few tiles around the city that will likely go your way in the near future, so you might keep it, but the city isn't that important.
Speaking of Mongolia, they have been a bit weird this game; Yes, as typical they were warmongers and invaded me even though I shared their religion (that war was how I got New Sarai and N. Hsia), but they also founded their own religion, and they have been a tech and culture leader the whole time. I do not have random leader personalities turned on.
An AI leader is less likely to declare war when pleased with you, however it can still happen especially if the AI thinks it has a bigger army than you (power rating graph F9-screen). Maybe a bad roll of the dice for you although you won the war so it wasn't a real problem.
I had to wage several wars against China to reduce them to their present pathetic state; The little salient down south (with Giza) was a recent conquest
Talking about conquest, I usually explore a lot more so that I know where my troops are going and how many AI-cities are left. Just 2 or 3 explorers can show you the terrain and the city placement of your bordering nations. At this stage, you don't have open borders with some of your neighbours, so it's useless to send some explorers there, so you need fewer explorers. Still, it's a good idea that you're building one at the moment. I would have done quite a bit sooner though.
I was thinking of going for a domination win, but as it is already the 1700s, I may not have the time
Domination is still very much possible. You have a good piece of land and are militarily stronger than your battered opponents. But you first have to get a decent economy and more powerful cities and that might take some time.
You have a good number of health and happiness resources. With the right buildings, the happiness cap and health cap will be above 20 in all of your cities and at the moment these caps aren't far from that (which is good). However, the average size of your cities is around 8 or something. That is pretty low in the 1700's. If your cities would be close to size 20, then your economy would be easily producing more than 4 times as much research (because costs don't increase that much with city size and momentarily your spending more than half your economy in maintenance).
The main reason that your cities are so small is that you're using few high food tiles and have created too few farms to be able to work many high food tiles. You need to take control of these workers and build a few extra farms and use them with your cities.
You also have a shortage of workers to change the land so that your cities can grow. Try to get a food surplus of at least 4-5 in cities with a granary so that they can grow quickly up to their maximum size.
Here is an example city of yours that will never become better if you don't do anything about it. The land is good enough, but it needs different terrain improvements.
City before improvement.JPG
I used the world builder to add a grocery to the city (for extra health) and added a lot of terrain improvements in a way I would be satisfied. The cottages of course don't start as towns, that takes time. But you can also see that you need quite some farms to be able to use the fairly dry land around the city. I also transferred one high food tile from a neighbouring city to this one but left another high food tile to another city as it can use it too.
City after improvement.JPG
I used the world builder just for demonstration purposes here. It of course takes quite some time to make these terrain improvements and grow the city. The city will become a lot better in the late game as biology means that you can change some of those farms into cottages or workshops as you see fit. And technologies like printing press and civics like free speech and universal suffrage will make those towns very powerful.
Hammer production went up from 5 to 14, commerce production from 14 to 41. A nice commerce city.
Just for fun and to show you what the late game civics can do, I used the world builder to give me the technologies printing press, democracy, liberalism and biology, switched to universal suffrage and free speech and used the extra food from biology to change some tiles from farms to towns.
City late game.JPG
The city now gives you 24 base hammers and 90 base commerce, pretty nice.
Note that the goal here was not to impress you with a high output city as I didn't add any multiplier buildings (forge, factory, bank, etc.). I just wanted to show that you need some more food to grow your cities and then the output from the land can become much much greater.