Well played, sir! After a somewhat rocky start, I think you have this game in the bag as well. Still, you did say you wanted to up your level for the challenge, didn't you?
The war wasn't really accomplishing anything at this point but making me unhappy, so I took what seemed like the most useful technology that he would give me.
Useful? Hmmm... Economics increases the Shield --> Wealth conversion-ratio from 4:1 to 2:1, but that's not helpful if you're still building infrastructure/units in most core-towns (which is where the effect will be noticeable); it makes no diffference to the Wealth-output of Specialist-Farms (which only get 1 shield -> 1 gold anyway). It also lets you build Smiths (if that hasn't been done already), which can be a hugely useful Wonder even if all you've built are Harbours and Markets (it also pays the maintenance for Banks and StockExes) —
provided you can afford to spend the shields on it. But right now you should probably rather be spending shields on Factories and Cavalries...
He wouldn't offer you SteamPower (or Medicine)? Either of those techs would have been my choice. Both are obligatory (unlike Econ), and Rails are super-important not just for increasing your core productivity, but also for increasing your striking-power and/or reducing military costs (because once you can move troops from one end of your empire to the other instantly, you don't need as many of them for defence, so you can commit more of them to any given campaign), and you need Med for SciMethod/ToE.
I switched many of my builds in those specialist cities at many points throughout these turns.
Be careful here! (Fully corrupt) towns destined to be pure Specialist-farms should usually get minimal/no infrastructure, because they usually can't pay the maintenance! So that means no Libs (increasing beaker output from 1.0 to '1.5' at 100% SCI%? Not worth it), no Ducts (1 gpt maintenance, then you have to deal with potential Happiness issues at Pop12, which may lead you to build a Market as well...), and
no Harbours unless there is Fish/Whales available (otherwise the Harbour will support more citizens, but not more Specialists).
So what is the corruption/waste like in Kutha and Nagsu? If it's, say, ~50-60% without a Courthouse, then you should be able to use it/them as 'real' cities, i.e. building a Courthouse (to speed subsequent builds), a Duct, a Lib, a Market, etc. (disband old units, e.g. non-elite Maces, for some extra shields into these builds) — but maybe
not the more expensive Industrial/ Modern buildings. Irrigated + railed Desert gives 3 food per turn, so they should quickly reach Pop12.
But if waste is already >75-80%, then even with a Courthouse, most building-investments will likely never pay off before the end of the game (this is a Small map after all). So while Kutha could use a Fishing-Harbour as a Farm, I wouldn't build a Duct there, and you'd be better off just building additional Workers/ Settlers/ Artillery (or Wealth) out of it and Nagsu.
Mursilis beats me to Electricity, but I am actually in a position where I can trade a technology to an AI. Miraculous! Giving up the incense stung a bit, but I thought it was worth it for a chance to get the wonder. I change my research to Scientific Method.
This was quite a good trade, I think, although including the Incense-export makes DoWing him problematic, since doing so will then make you a deal-breaker. But if he DoWs you, you get to keep the Incense — and your trade-rep!
But I hope you made this trade on your turn rather than his, and also traded Sanitation to anyone else who could afford it, so that Mursi couldn't profit from your hard work?
Now... I don't know how tight this race is going to be, but it seems like Plan B is superior. Korea and the Hittites both have Scientific Method and may be going for it already.
Just because they have the tech, does not mean they have started on the Wonder. You can check this by using the F7 (Wonder) screen.
And if they haven't started on it already (which they might not have, if at war — and they can't, if they've Mobilised), then you might well have a good chance at getting it yourself.
I don't know if the AI will prioritize this wonder, and how quickly they could complete it. Should I do an investigate city espionage action in their capital to get that info?
Not necessarily in their capital, unless you
know that's where they're building it (and note that the AI does not always make rational/good decisions about where it starts building its Wonders!).
If/When they do start on it (or move the project to a different town), you can get a pop-up telling you so (you need to have the 'Show Wonder initiation pop-up' checkbox set in your Game Preferences: CTRL-P screen), and then you can use F7 to find out which town they're building it in. Provided you have a reasonably up-to-date Territory Map, and a good idea of their tech-progress, though, you still don't
necessarily need to investigate the city building it.
If you know how big a Wonder-building town is, then by examining the tiles around it, checking its distance from their capital (to estimate waste), and taking the potential boost from a Factory and/or CoalPlant into account (can you see Pollution?), you can make an educated guess how many shields that town is getting, and therefore how many turns it will take them to build the Wonder, from the date they started it (assuming they didn't switch to it from another magaproject-in-progress, such as UniSuff or BattlefieldMed).
If even your worst-case scenario (highest shield-tiles all being worked, no waste, Factory + CoalPlant already built) still puts them (far) behind your schedule, then you probably don't need to pay to investigate the town. If you're still not sure, then yeah, do it. Investigating cities is a Diplo-job, so it's relatively inexpensive, and doesn't upset anyone. But you
can't do it while at war — then you have use a Spy instead (which means you have to build the Intel Agency, and pay to plant the Spy first — which can itself cause war, if you're caught!)