Some serious coaching needed on cultural victories

They are helpful, but some games they are not readily available. They are good to cut a few turns, but I don't think necessary. Maritime are just as useful. Mercantile, while not netting as much culture as pure city-states, can still potentially give you 10+ happiness, which translates into 5+ culture with policy. If culture city-states are not near, there should be useful alternatives.
Yes, Maritime and Mercantile can be good too.

Keep an eye on the resources available from the Mercantile though. With the latest patch, you don't get any extra happiness from them when going from Friends to Allies (instead you get the luxuries). It could be the case that you already have access to the luxuries that would be provided by a Mercantile CS, so in these cases, you can just stick with being Friends for the +3 happiness.

If you have the time, then go for them. Why not? Else, I'd cut Stonehenge. Keep watch for religious city-state quests. Even just friends should be enough faith to get a religion up fairly quickly.
You are right that Religious CS can make up for the faith bonus of Stonehenge.

However, an early Stonehenge does mean that you are likely to get an early Great Engineer, which can be used to rush a later wonder that you really want.

1. Manually build monument, so Legalism gives you a free amphitheater.
Yes, that is right.

The main scenario where I will want a free Legalism Monument in my capital is if my capital is in a prime desert location and I plan to get Petra (which also provides a free Amphitheater). So in this case I'd rather have a free Monument via Legalism.
 
Well if you build Petra and Monument you can use Legalism for Opera House - which is unlikely to be hard build by capital since you want to work the Sistine Chapel first. Also I wouldn't recommend building a Monument unless you settle on a hill or something. You need that production for something else. If you can't build it under 8 turns, you're likely wasting your time when you should be making a Worker/Granary for growth if you're going down Tradition.

As for other pointers:
-Sometime you just don't have a good start. You can turtle it out but be prepared to adjust to the game. Sometimes sub300 is just NOT possible giving bad starts. 300-320 is still good since most AI can't win by that point (I say most because runaway civs are likely to outdo this).

-I generally forsake all wonders in ancient/classical except for Oracle if I have bad production. ToA/HG is ALWAYS a gamble on immortal. There is no guarantees at all. I've seen ToA get made BEFORE GL or as late as T91. It really is variable. And when you're close to an aggro civ you do NOT want to take a gamble by wasting 18+ turns for ToA/HG when you can be building 4-5 archers for defense. Then again if it's an Emperor game you generally have a bigger window/chance to snatching ToA at least provided you went to that tech relatively fast.

-Liberty can be an alternative start provided you have GOOD growth but poor production. You can generally get mileage out of the free worker + settler + GP than overgrowing your cities. If I'm Montezuma or Siam I usually give Liberty a chance.
 
Well if you build Petra and Monument you can use Legalism for Opera House - which is unlikely to be hard build by capital since you want to work the Sistine Chapel first. Also I wouldn't recommend building a Monument unless you settle on a hill or something. You need that production for something else. If you can't build it under 8 turns, you're likely wasting your time when you should be making a Worker/Granary for growth if you're going down Tradition.
Legalism for Opera House is an option, but it really depends on your strategy. If you're mainly going down Liberty or Honor then yes, you can do it. The main issue is that delaying Legalism means you can't get Landed Elite or Monarchy (or finish the Tradition tree) until after you have built Petra. That can be a big issue if you were planning to stay small and finish Tradition quickly.

-I generally forsake all wonders in ancient/classical except for Oracle if I have bad production. ToA/HG is ALWAYS a gamble on immortal. There is no guarantees at all. I've seen ToA get made BEFORE GL or as late as T91. It really is variable. And when you're close to an aggro civ you do NOT want to take a gamble by wasting 18+ turns for ToA/HG when you can be building 4-5 archers for defense. Then again if it's an Emperor game you generally have a bigger window/chance to snatching ToA at least provided you went to that tech relatively fast.
In the early game, it helps to keep an eye out on what wonders the other civs are building. You can also often indirectly infer which technologies they have access to.
 
Also I wouldn't recommend building a Monument unless you settle on a hill or something.

I usually restart the map if I can't get a 2 hammer settlement. 7 turns for scout and like ~16 for worker? No thanks.

Been playing around with monument. I had always kept it til later, relying on the Tradition opener and (with luck) ruins, but I've been liking the early culture. It is only a handful of turns for both monument and amphitheater, but getting out both seems to help me finish Tradition before moving on to Piety. There are a few that suggest going to Piety right when it opens, but I just can't do it. I don't see how putting off growth percentage bonus and aqueducts is worth it.

Edit: I should also point out that this is a good counter-argument for my comment of Oracle not being that useful. That wonder does more or less remove the "Should I finish Tradition then go to Piety?" issue all together.
 
Getting piety before Tradition finisher is lacking. Also you're unlikely to get 3-4 world wonders anyways that early in the game and Reformation gets more benefit if you build CI before it.

It takes like 18 turns to make a worker if none of my tiles has a production aside from capital and palace base, which blows. I generally don't want to respawn maps but this just begs for a slow start. It's funny how something so small as just 1 more hammer that could have such influences for the rest of the game.

Oh, and I also tend to forget about monument even if I have good production but very close to another aggro civ; which can be frequent in fractal map settings. Get Mongolia/Huns/Greece, and I know I'm in one hell of a bloodbath early in the morning.
 
Yeah, I generally finish Tradition before starting on Piety.

It takes awhile to get setup for Reformation. Also, if you wait a bit, you can sometimes finish Chichen Itza before getting it.

The main reason why I might consider opening Piety first is if I am rolling in Happiness and I can significantly boost my cultural output with Mandate of Heaven. In such a case, I'd consider opening Piety and getting Mandate of Heaven before going back and finishing Tradition. But you are delaying the Tradition finisher by two SP's, so that is a significant opportunity cost.
 
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