Round 4, 580 AD - 940 AD (18 turns):
A short set since it's been a couple of days since the last update and a lot happened.
580: Paper -> Optics. I decided to trade Paper around so I left research to get a turn's worth into Optics.
I began the round with opening borders with Churchill and Pacal and found that I could indeed trade with both, granting +32 beakers at 100%. I found this very strange, partly because I had not encountered Pacal's borders and partly because it seemed Hannibal blocked both of them from me (more on that later). At least Churchill cost me nothing since I discovered him this very turn but I might have missed out on a little trade with Pacal.
600: Optics -> Education, going for Lib. I trade Paper around, starting with Gandhi since Pacal was his worst enemy but no other conflicts threatened. All in all I gave Paper and got a -1 relation from Gandhi (after trading with Pacal) in exchange for getting Currency, Monotheism, Optics, Construction, 55 gold and a map. Pretty good all things considered.
Here's a look at the strange circumstances of my trade with Churchill and Pacal. I can't figure it out.
a) I hadn't discovered Pacal's border but one of his cities is touching Gandhi's borders, is that enough? Note that it is cut off from his main cities by Hannibal.
b) I can't reach Churchill without going through Hannibal's area, what gives?
An overview of the northern lands, Hannibal is the largest with 9 cities, followed by Churchill at 6 and Pacal with only 4. In total 19 cities in the hindu block to Gandhi's (at this point).
620: A trireme finished in Den Haag and I upgraded him to a caravel for 50 gold to start looking for the last two civs and to race for the cirumnavigation bonus.
660: Pacal becomes a vassal of Hannibal peacefully, making Hannibal and Gandhi each other's new worst enemies (good since I was H's worst enemy up until now).
680: My caravel doesn't have to search long before I find Darius on a landmass just to the east of me. He's quite backwards meaning we're at about tech parity, though not the same tech exactly. He only has 3 cities though and is tao but not the founder.
I decide to not trade with Darius immediately, he's only made contact with Pacal and Hannibal who're both annoyed with him and thus wont trade techs so I decide to wait and see. Since Darius is taoist but not the founder, he must be semi-isolated with the last civ, who's in all likelihood going to be the taoist founder and on friendly terms, though you never know with aggressive civs.
700: I decide to go ahead and trade after all since Darius is pursuing Optics on his own. I do this despite there being a chance of me invading him later since I can use the techs he offer for my military and I don't want to completely fall behind the others.
720: I was hoping for a GS to speed Education along to race for Lib but 79% wasn't enough this time and I get a GM. He can help me achieve the same goal though by making a trade mission and letting me deficit research for a while. I decide to do a suboptimal trade with Gandhi rather than waiting for a caravel to be built to take me to the last civ who built the TOA (the one caravel I have being concerned with circumnavigating). I want the money fast, the exact amount is less important. The GM starts his 6 turn trech toward Bombay (the thrice holy city).
Darius becomes WHEOOHRN but I find out the next turn that Wang and he is friendly with each other and he hasn't got astronomy to reach anyone else, what is going on?
740: UoS BIDL (Churchill). I meet the last civ, Wang Kon who is indeed taoist and on friendly terms with Darius. I waste no time trading with him as well, putting the three of us on tech parity. Wang Kong has 6 cities making it 9 total on the eastern landmass.
The most important thing here might actually be the map believe it or not, the rest is just gravy. I want that circumnavigation bonus if I'm going to war in the future. Since Darius and Wang are the weakest targets it could also be very useful to know the lay of the land should I invade them.
Their land is not the best but it's great compared to my joke of a land. Then again they're two civs sharing it. The food heavy Persian capital looks juicy and has TGW, THG and AW, Seoul has MoM and TOA and there's a taoist shrine in Wonsan. A couple of other cities looks decent for both emancipation cottages and/or industrial-era production.
760: Hagia Sophia BIDL (Pacal). I get the circumnavigation bonus from Wang's map.
780: Gandhi temporarilly becomes pleased with me and I beg 40 gold from him. Churchill got the same treatment a few turns earlier but refused. It's likely I hadn't known him long enough to get 40, I should probably have waited a bit longer or begged for 30 gold.
800: I'm asked if I want to stop the war against Gandhi. What war? Well I guess Hannibal declared on him then, it would really help if we were told who's fighting when we're asked to make these votes. I can only imagine how difficult this might be to keep track of in an earth 18-civs scenario for example, especially when you've not yet been told because it happened the same turn!
I vote no of course, not that it's likely to help.
820: The vote is in and unsurprisingly, Gandhi gets his wish and the war is aborted before it even begun.
840: Education -> Liberalism. Gandhi and Churchill get Astronomy, Pacal follow a turn or so later. My GM arrives in Bombay the same turn and I cash in 1100 gold to keep the slider at 100% all the way through lib. I would probably have gotten a couple of hundred extra in the twice-holy city of Vijayanagara that is on Hannibal's continent but it would have taken 3 turns extra even though I had a galley ready to transport, which would cost me a turn on Lib since I lack funds.
860: Hannibal becomes cautious and I OB with him and trade resources. He's WHEOOHRN again though which is not good. Pacal is his vassal and he has a peace treaty with Gandhi, making me and Churchill the only possible targets. He's cautious with me and pleased with Churchill, which means, you guessed it, I'm probably his target!

Fortunately his closed borders policy with Gandhi won't let him send troops until he gets astronomy but this is not good for diplo and could be dangerous after he gets astronomy.
940: One turn from Lib and I end the round. Taj Mahal BIDL (Hannibal). Sistine Chapel BIDL (Pacal). I belatedly switch all EPs to Gandhi deeming them insufficient to carry out city revolt missions on Darius or Wang especially since I've not been able to spie on them very long. Gandhi is closest for stealing purposes and also the most dangerous opponent.
Close to the end of the 4th round I find an unoccupied (apart from barbs) landmass just to the north, it looks sizeable and green. Churchill, Pacal and Gandhi already have Astronomy though so if I plan on claiming it I will have to race them.
Empire shots, all cities but silver-junk city is building universities to enable Oxford's in the capital. I'm trading stone from Churchill which should help in that endevour.
Techs, I can trade Gandhi or Hannibal but either will piss off some civs. Hannibal will give Nationalism for Edu+ 350g or Guilds and Music for Edu+ 130 g. Gandhi will give PP for Edu and 485 gold.
Wang and Churchill are close to getting Edu on their own.
Demographics:
Relations:
Hannibal is the worst enemy of Gandhi, Wang and Darius.
Gandhi is the worst enemy of Hannibal.
Wang is the worst enemy of Pacal.
Trading with H is going to piss off Gandhi and the taoist block and trading with Gandhi is going to piss of Hannibal (and in extension his vassal Pacal).
I'm thinking cuirassiers as closest military tech since I have both horses and iron, note that Hannibal and Pacal already have it. If I trade H for Nationalism I might then be able to trade him Astronomy (that I'll pick from lib) for Military Trad. OTOH if Hannibal is looking to invade me, maybe I shouldn't give him Astronomy.
With the green land to the north I'm thinking I should try to claim that asap, so settlers, workers, units and transports are needed once the universities have been whipped, the capital should have some time for it after finishing the university there before the other cities have enabled Oxford's. If that goes well I can then consider a cavalry invasion of wang and darius or perhaps get cannons there as well.
In any case I have a shot at expanding here both by peaceful and by agressive means and I think I should do all I can to get some more land!