You can take Liberty free settler to settle your second city and then build the NC. Then expand some more, to 4 cities. When you gain more happiness (Notre Dame is advisable) settle the rest. AI's cities count too, of course. But they should be good cities, e.g. capitals. Which can be annexed. If you can't/don't want to capture capitals, settle you own cities. After... I don't know, maybe 150 there is no really point to settle more, just capture some. That's the general path I think you should start with. For early science finish you should build 4 cities before NC, but it's not easy to do without delaying it and takes a lot of practicing.
As for your saves:
The first one: I'm exceptionally surprised by your policy choices. Don't do that. Start with Liberty and finish Liberty. As smallfish suggested, pick great Engineer to rush Porcelain Tower. Build Hagia Sophia and get another engineer to rush Notre Dame. Mixing up policies like that is the worst possible choice one can make. Later, when you enter Renaissance, unlock Rationalism and finish at least the left side. Scientific Revolution gives you two free techs, so save it to the end to bulb the most expensive ones.
You don't need walls. It's better to have an additional archer per city. Build monuments first in all of your cities. Your culture is very low for this stage of the game. Try to ally cultural CS, build temples. Build granaries as well, they help growth. Let AI build the Pyramids and take it later if you want. Building settlers instead would have been better choice. Because you do need more cities. At least one to the north by spices and another one near marble. Bad city spot but this map lacks options. Akkad should have been built one East of its current location. By settling where you did you lost both river access and coast access. Killing Siam is a feasible way to expand on this non spectacular map too. Probably the best way.
I noticed you never traded with anyone. Sell spare open borders, luxury resources and unused strategic resources to AI. Friendly AI will give you 240
per lux, 50
for OB and 45
per strategic. Right now you can trade Washington 4 horses and OB for 230. If AI don't have enough money, take whatever they have plus as much gpt as you can get. If they dislike you, they'll pay less. Rami will buy wine and OB for 51 gold and 2gpt. Not much but better than nothing.
One more thing on diplo: the request not to settle near you is pointless. It does nothing other than aggravates AI. Don't denounce them as well, unless someone you're interested to be friends with did the same. Global politics screen doesn't show America denounce Siam, so there was no reason for you to do it either. Let him denounce you if he wants.
You should also check the demographics screen often. You're last in soldiers and Siam is not a big fan of yours. I suspect he's going to DoW you soon. So it's a good time to start building some army and take his cities.
I'll peek at your second save and post in a bit.
Was a revelation to me at the time. 
), RA's would be hard to pull off due to AI's bankruptcy and backwardness. You need lots and lots of cash to fund them. You pay for both parties in addition to penalties for being eras ahead. Therefore I'd definitely recommend having as many cities (more cities = more universities = more GS + more science) as happiness can support and as powerful economy as possible.
at turn 200 - I don't think I've ever seen something like this before. 
He visits the forum. Come back, Wainy! 
