Well this will be my submission for this gauntlet, it is a truly great result for me, and I cannot see how I could have done it faster, thinking back about every move, I can see maybe 1 turn possibly 2 if I had accelerated the war in a couple of occasions where I just waited for an unit to heal and not run risks, but nothing else, the rest I played optimally. The only true mistake, which would have saved me a 2-3 turns, is when I settled in t1 (moved my settler), I should have settled on top of one of the salts instead of settling in the plains. Misclicked.
Culture:
1 culture ruin. I had 5 cultural CSs, and I was influential to varying degrees with 4 of them, throughout the game I spent I think 1500 gold, but could do some barb quests, mostly camps, so that saved me some money. I was unlucky that 1 barb camp for one of them was unreachable (completely surrounded by mountains, unbelievable). Attila was the laggard, not sure when he built the monument, but he would have reached 300 by t125, so I guess t38-39, normal time. There were a few other later monuments but they all popped ruins or were too far away. I closed Liberty t113, but could have closed it earlier, but preferred to save the money just in case, as I saw that the war was going to be the limiting factor. I bombed Attila on t118 or 119, don't remember exactly.
The war:
I built a scout and then rush bought another. After monument, I built a first army of archers and paired them with 1 scout and the promoted warrior to spearman (ruin). Those guys went directly north to kill Morocco (t78), and then went into the woods in the west to kill 2 AIs there, Venice (t92) and Ethiopia (t112). They finished their job 8 turns before the win.
The second army (CAs) went north and west to kill first Arabia (t102), then Delhi (t112), and finally Carthage (t120 win). Would have been faster to go directly west, then north and come back, but I had the feeling that Harun was building a settler, and did not have any units to spare to leave there and grab it, so I went and killed him first. I think 1 turn difference anyway so no big deal.
The luck:
I had the minimum luck necessary for the 115-125 range. 1 culture ruin, 1 tech ruin (Minery, unfortunately I was only 5 turns away from finishing researching it, but it was key anyway), 1 pop ruin, 1 promo ruin, ideally the warrior to spearman, and the laggards being within 10 turns. The rest of the maps were gold or maps. You usually have 3-4 out of the 7 within the 10 turn tange, and being a laggard depends on them not hitting culture ruin, more than the monument timing, so it helps that you are of course aggressively scouting around them to steal those ruins and just messing with their little mechanical minds.
If I had a second culture ruin take 3 turns off, or an earlier useful tech (archery or mining is ideal, the wheel or AH is also cool, but prefer one of the other 2), that is another 2-3 turns. I don't count the start here as for record time, it is clear that you have to roll a 2 mining luxury start that has at least another 3-4 2-1 tiles (2-3 if the lux is salt). These starts are not difficult to get, and remember in Great Plains it is very likely horses will appear if you are on the plains.
Tips:
I cannot stress enough how important are the first 35 turns. I mean you have to micro manage every single turn/every single unit move, but those first 30 are essential. The best advice so far is Vadalaz's, rush buy that second scout asap. The initial exploration scouting towards your neighbors, first trades timing are absolutely essential (99+whatever you can get by offering 1gpt), the careful timing to get the maximum return, the calculations for the DoW/Peace deal timing. Calculate everything in those 35 turns, they will dictate the rest of your game.
On to the diplo.
EDIT: with screenshots now