Zong,
Disclaimer -- I'm no expert. I've never played at Diety, though I seldom lose Emperor. I only recently came back to Civ3 Vanilla, so take this for what its worth.
First, is this a hyper-resource mod? Or a scenario? Or do you just have insane luck? You've got 6 different luxuries within 8 tiles of your start location! And not just you. The AI seems to have lux out the wazoo, too. Both bonus resources and strategic resources are well above norm, at least in my experience. Maybe its something to do with playing at Regent level, and though I've played a lot of games since playing at Regent, I think I'd recall something like this. I'm inclined to think that bonus resources this extensive will be to your disadvantage. The AI makes bad decisions on improvements, sure, but its hard to go wrong whatever you do with cows. (Keep it clean -- this is a family site.) Your big advantage is in wisely choosing improvements, and hyper-resources diminishes that advantage. Worse, it will teach you sloppy habits. You won't learn early game cold, and its going to make moving to higher difficulty much harder than it need be.
Anyway, I'll comment assuming its just standard, vanilla C3. Looks to me as though probably the issue is a matter of a slow start because of a couple improper improvements early on.
I see you mined the wheat N of Beijing. You have a plains/hills start location, which is about as weak a start as you can get by with. You WILL get the benefit of the irrigation of wheat under despotism, so you should do that.
Wealth is almost never useful in the early game, or even in the later game. Tundra, or desert you cannot irrigate, that's about it. Very rarely as something to do while you wait for a town to grow or a tech to finish, though a prebuild is almost always a better choice. If you are ever tempted to set it on wealth, getting only 1 gold per 8 shields, ask yourself if there's really NOTHING better to do. Unless there was some vital strategic value in that location, you would have been better off putting that town elsewhere, or not producing the settler at all -- leaving him as 2 tax-men would have been more profitable, and would have freed up the 30 shields for something useful.
You need lots more workers. Nine towns should have at least 13 workers, and I'd prefer closer to 20 because there's so much jungle to clear and mountains to develop. That said, I would not be clearing jungle or mining mountains just yet. Not until your useful towns are developed. You're not going to be able to build many workers without irrigation of your bonus food tiles and conscious development of a worker farm. At a guess, you probably don't need to switch to mountains/jungle until maybe 1000AD at the rate you are going, and in a town like Shanghai, not until after rails.
Its counter-intuitive, I know, but granaries are almost always a waste. For those 60 shields you spent at Shanghai, you could buy 3 chariots/spearmen/catapults/archers, 2 horsies/settlers, or 6 warriors/workers. China could build a barracks AND two veteran warriors. In a town like Shanghai, even after you get out of Despotism, you will rarely produce workers, and will probably never turn out a settler. There's just not enough food, at least until you get rails. You would benefit from having a granary in Shanghai maybe 20 times the whole game, if that, and its going to cost you 1 GPT for the rest of the game. That's not a very good return on investment. (Totally corrupt outlying towns will produce a worker every 10 turns, so it would be silly to make a worker out of a citizen who WAS producing shields when you could make a worker out of a citizen who was not.) Further, 10 food is kind of awkward for all except a 1, 2 or 5 excess food town. A good-looking despotism town might accumulate food at 3/turn or 4/turn, neither of which are probably good candidates for a granary. The 4/turn means 3 turns for pop growth with the granary, 5 without. Even excluding the maintenance costs, the opportunity cost of shields makes it a losing proposition -- you could have founded two new towns for the same expense and come out well ahead. Alternatively, you could have alternated military and worker, which would be far better unless you have a really good reason for needing the workers in a hurry. (I'm convinced that at Emperor, such a situation, needing workers NOW, at the tail end of the Ancient Age, is usually a pretty good sign that I screwed up earlier.) A 5 excess food town gains no benefit at less than 4 shields -- it would take 4 turns to get enough shields to get a worker anyway, and it will only speed up worker production by one turn at 4 shields. Settlers are even worse. Bottom line -- go ahead and build a granary if the town produces 5 excess food AND 5 shields AND you are planning to produce at least 14 workers or 7 settlers from that town before you let it grow to be a production center. That's about break even; not necessary, but not a bad idea right out of the box. The rest all have to be thought through on cost-benefit. Depending on the map, you may find yourself in a situation where you win the game having built only two or three granaries.
Kind of a small thing, but you have GOT to get that Elite spearman out of Nanking. Fortified in all that jungle is just asking to have him killed by disease. Send in a regular spearman to take his place. Better yet, you could simply fortify him on the hill SW of Nanking on the incense. Jungle will prevent anyone from attacking you by surprise, so you can always move back to town if someone moves towards Nanking. Maybe disease isn't as bad at lower difficulty. I don't recall. But needlessly risking an elite seems suboptimal.
I see you founded Xinjian on gems. Costs you a few GPT, but what the heck. However, if you are not concerned about building on gems, you should have built on the gems next to it. Then the borders would have merged with those of Shanghai, and you wouldn't have had to build culture and wait for it to expand to claim all that space. That's important. You need to close off all the open land as early as you can to keep settler/spearman combos from walking through and settling "your" lands.
Speaking of Shanghai, there's no point in it being where it is. One tile SW, on the hills, would have been better since you would have immediately claimed the spice. Alternatively, had you placed Shanghai right on top of the spices, not only would you have had the deer immediately, Gandhi wouldn't have put Bangalore where he did. Maybe he would have encroached on Persia, or headed south or something. I don't know. But he would not place a town where he had 8 tiles under Chinese cultural influence if there were any other place to build, particularly not with Beijing so close. (I should qualify that with I don't know where Delhi is. It might be that size 7+ city SW of Karachi, but even so, I'm not sure I've seen AI purposely try to pull a culture flip. And a flip is easy enough to prevent with a few MPs.)
Generally, once you find grassland, I think its best to explore and settle it until it runs out. Grassland is just too good to leave idle. There's grass east of Beijing. That's your prime expansion.