Looking back at the game though I'm a little surprised you got the fail gold on ToA. Outside of stonehenge, the great wall, and the mids the ai civs don't seem to be prioritizing wonders much, and in your game nobody even started working on the mids yet, Bismarck had actually just started the GLH that turn.
Yes, failgolding also works better on higher levels when the AI actually builds wonders. The cheaper the wonder, the more likely the AI is to start building it early. They are more likely to pick a build the fewer turns it would take them to complete it. Stonehenge and Great Wall are usually the only safe bets for failgold that you know will be built very early. ToA is my next favorite after that, because it is usually the first marble wonder that is built (except Oracle, but that one is better to build yourself if you can). ToA is also a wonder I never want for myself. Failgolding mids is very risky, on monarch it can vary so much. Someone might build them 700BC, or it might be nobody gets them up until 500AD. They are so expensive that it takes a lucky RNG roll for an AI to start building it, especially if the AI doesn't have stone.
How many more cities did you settle before you went to war? Other than the 6th city you had planned I saw a nice riverside grassland tile that would grab the wine and cows, and there's another coastal city spot that stands out to me that could grab those pigs and the fish. Neither seem like they would be very good for cranking out units though. I would probably settle the horses and cow up north, the cows and wine in the east and go at it with 7 cities.
I settled Horse+Incense+Clam northwest and double fish+pigs on the island south of pigs. The island city wasn't that useful, but it built triremes to fend off barb galleys, so at least some use... In a longer game it would have been a lot better. Great GP farm that could have worked 7 specialists in Caste/Paci without starving (or more while starving). Would have been great for pumping cuirs as well, food>hammers when it comes to production because of the whip, but I couldn't bother with the hazzle of shipping cuirs to shore from there...
Either way getting across that map and conquering every city in 20 turns seems like it's not the easiest feat. I would imagine you didn't need catapults.
No no, not conquer every city. Take capitulation as soon as possible and then gift them their cities back. The only city I kept was Moscow because it had the Pyramids (for Police State). Cathy for example, I declared 600AD, took one city that turn. Next turn took Moscow+one more city and she capitulated, so war was over the next turn after I declared.
Absolutely no catapults. The point with mounted warfare is being fast. In this case only 2 AI had Feudalism, so odds where mostly in the 90%+ range even against their top defenders. Longbows aren't a problem either. You might have losing odds if they are on a hill with heavy cultural defenses, but some losses are always expected in war. In total I lost 11 cuirs while conquering the world (built 49).
War strategy for final cuir war is quite straight forward. Set all cities to produce nothing but cuirs. Check them every turn if they can 2 pop whip. If yes, check if it can still next turn with better overflow. If no, whip. Then sometimes some minor micro adjustments to see that they don't go over 54

the first turn of putting hammers into a cuir (1 pop=45 hammers with forge+PS). By the end of the game all of my core cities were looking at 30-50 turns of whip anger.
Edit again: So I just realized you settled on the dye which changes everything, and I can see why, the extra commerce and happiness is a huge bonus early. It also opens up a riverside tile, and now I'm starting to see a case for pottery before bw. By going AH, Fishing, (popped a map the second time) Wheel, Pottery, Mining, Bw I can go worker, work boat, work boat, quecha, settler. I haven't tried playing it that way yet but I'm pretty sure I can build the settler on size 4 working a cottage and have a build a road to city 2.
Yes, settling on the dye is the key. You don't get happiness until calendar, but a 3c city tile for capital is a huge early boost. If you don't settle on the dye you should probably cottage it early.
I built settler at pop 3 working cows+double seafood. Finished BW just before settler was out so that I could revolt to slavery while settler is moving. This is often the best time to revolt, as the anarchy has no effect on your second city (doesn't delay production of settler and you are back in business by the time second city is settled). Tech order is the one you described. I did get Hunting from a hut though one turn before I completed AH, which would have given a 20% boost to my AH beakers that last turn, so maybe 2-3 extra beakers from that.
I didn't tech masonry or hook up marble before building the Oracle btw. Oracle is so cheap even without marble that teching masonry would only delay it. If you are Oracling CS, there's lots of time to get the Oracle ready for completion between PH and CoL.