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I do not have the answers to your questions, though. I have no idea when you should be producing them.
Having them oversee a resource drop off building increases the resources dropped off by 20%.
Yup, only really worth it for gold mining.
Generally, you want to build them pretty early on, if you can afford to. You generally want 1 for your opening build order. As base Chinese, you probably want them after Song Dynasty since you can just build them from the landmark.
As Zhu Xhi, generally if you've got like 7-8 on a resource, you probably want an IO. And you usually want one going around collecting taxes, so for instance in feudal if you have like 25 vills, you might have say 12 on food, 10 wood, 3 on gold, and 3 IOs (1 wood, 1 food, 1 tax and potentially supervising upgrades).
And no you shouldn't wait for 15 vills. If you could build 2 vills for the price of 3 and the time of 1 in aoe2, you'd do it right? Well you'd definitely do it in aoe4. So at 10 vills on a resource, you definitely want an IO. Its just debatable what point you want one before then.
There's no specific amount they bring in late game cause it varies based on where your buildings are and how much you're producing, and what you're producing (its 4 tax per unit regardless of the unit, so producing spearmen will generate a lot more tax than producing knights or bombards). But generally you can expect somewhere in the range of 300gpm from tax alone in imp. Maybe more maybe less depending. If you don't have your buildings close to a TC (or other tax dropoff point), then generally they just won't collect from that building, so if your production and/or resource gathering buildings are far away you'll get less.
And yes, it will tend to be less than English farms, or having a bunch of traders.
If you don't agree, I look forward to you hitting rank 1 since you know better than every high level player.
10 Farmers with perfectly placed mills and without the 100% bonus landmark were dropping off 37-43 GPM (I turned off all other eco while I did this to ensure it would be accurate. So that averages to about 4GPM per farmer which is 1/3rd of Britons 12 GPM per farm.
10 Farmers with perfectly placed mills around the landmark straight up doubled to 8 GPM (74-86 GPM).
10 Farmers with granaries and the landmark was harder to calculate, but it did seem to be about 20% more gold, but if fluctuated a lot more given overlapping granaries (gold is dropped off every time food is deposited). Don't ask me to calculate what cramming 50 farms by granaries will do because the map wasn't set up for that - dropping ideally placed TCS for taxes would have taken way too much room here.
But I think it's fair to say they can get about 3/5ths of what the Britons do in the late-game, somewhere near 9ish GPM per farmer if it's laid out correctly.