Some minor advice.
Since you are playing on Prince, you can't rely on stealing workers early, so you need to build one yourself. I see you are teching in order to get the improvements for the workers, but getting a worker at T30 might be late. You are already at population 3 and could build a settler. I would add a worker faster, and since you got a culture ruin, you can skip monument, you get a free one with legalism (with only a minor delay to culture), so you will benefit a lot more from a worker.
The tech order for this map, for me, would be pottery, mining, animal husbandry, and then calendar, and so on. Maybe add archery if barbs are giving you trouble, and add an archer to your early build order.
Mining is very important since it allows you to chop forests, this adds hammers to your capital and greatly speeds up buildings. First thing to do with a worker is chop the wood east of Beijing, this will add 20 hammers (i think), and will speed up the current build. Then start chopping the forests on the citrus, you will need to chop them anyway for improving them with a plantation. Chopping further away from the city gives less hammers, and even less if the tile is not inside your cultural borders, but it is worth it anyway. Those early hammers accelerate you a lot.
You could add a second worker to your build orders since there is so much work to do, this would be built faster with the chopping. One worker could start on pastures while the other chops the forests around Beijing. Try to improve all the tiles the city is working: pastures on the cows, sheep, add a mine on those hills (this will help with settler production). After that connect the luxuries, if the AI has money (at least 7GPT) you could improve luxuries faster and sell them for 7GPT, be sure to keep some happiness for yourself, not to go into unhappiness, but that would be unlikely this early.
You have good land for Pantheon beliefs so include a shrine in the Build Order. This is why pottery is important, you can build a shrine as soon as you research Pottery, or you could finish the current project and then start work on the shrine. The earlier the shrine the better, since you get a pantheon faster, and start adding faith for the first prophet.
You have a few good options: the one that gives culture from pastures seems very good, since you will be working both cows and sheep so that is an early +3 culture. The expos seem to have a decent amount of pastures so they will also add culture.
There is the one that gives +1 food from citrus, banana and wheat. With all that citrus around it is also a good option to accelerate growth.
There is also the pantheon that gives culture and faith form incense but I see only 3 in the area and 2 of them are on flat desert, so you would not want to work those tiles anyway.
With a fast expo in the west, desert folklore can give you a lot of faith generation since you have some sheep on desert, floodplains and other desert hills. This will give you a very powerful religion and let you choose religious building which can be very good. So depending on how fast you get your cities up, this can be very good. The capital only gets +1 faith, so very early the boost is not that big.
So getting a pantheon fast is a good option. Of course at T9 you did not have all this information, but shrine would have been a good idea for the culture from pastures anyway.
My initial build order would have been: scout, worker, shrine, worker, (scout), settler, settler, granary, archer, archer. Maybe add the first archer sooner if there are too many barbs. Once getting the workers up, return the warrior home for protection, and use the scout to see potential city spots. The archer is useful to get barb camps and get city quests.
For the expos, I would try to explore the dark area east of Beijing, that river seems nice with the horses, bison, and maybe citrus in the 3rd ring. If the land is good, that would be my first expo.
The one you marked is also good, it can be a good city with Petra, a decent city without it. But I would plant it either one tile east of the mark, on the hill or two tiles, on the citrus. I don't think you can, since it is too close to the CS.
For the city on the hill the advantages are: 1 extra hammer and better defense for being on hill, also adding incense in the 3rd ring.
The city on the citrus has other advantages as well, it is next to a mountains so it can build observatories, it has more land since you are further away from the CS. Also it will always work the citrus so that is 1 extra gpt for the entire game (you also lose 1 gpt from not being able to improve the citrus so this is a tradeoff). But the big advantages come with Petra, it has 2 sheep in range, and one river hill. Those are 3 food 3 production tiles with Petra, you also add cow in the 3rd ring (3 food 1 production, two with stable), and various hills which will be 1 food 4 production tiles. There will be some overlap with Beijing but that is not that important. You also lose access to stone, so and also to Stoneworks.
So both spots have it's tradeoffs. I would probably chose the mountain spot.
For the 4th city I would check the area with the salt between Beijing and the American lands, maybe it's good. Another decent spot is SE of Beijing, on the tundra hill next to the mountain. You get 2 deer, stone, dies, and some river plains. Of course this all depends on other expo placement since you don't want too much overlap.
A city on the coast could help, since you can use cargo ships for internal trade routes to/from the capital. See if the coast SE of Beijing is worth settling, with some sea resources, and the deer already visible, it might make a good city.
I'd play the first 60-80 turns.
