... I figure either I'm missing something important or the average is a little off, because an average should occur more frequently that this.
Sometimes the map is the problem (too arid, for example) Are you playing on new random maps each time? If so, I guess you are missing something. 5 or 6 cities in 2200BC should not be such a rare thing.
My first normally moves one space then settles, because the initial position is usually decent (and I prefer an empty grassland to a shield under my city, because both have 1 resource shield when settled). But the other often finds a mass of either grassland (no specials) or poor land the first place he goes, so he might spend 6-8 turns looking for a site.
Your 2nd settler may be part of the problem. I try to settle mostly on grass, usually
the first possible grass tile. I am happy with that, if there are 1 or 2 forest tiles in sight (for the size two production boost), even without a special. I'd prefer that each city have at least one special, and am willing to spend maybe an extra 1-2 turns for a good visible one, but not 6-8 turns searching in the dark.
Consider how important one turn of production is later in the game. If you waste a turn early on, you've lost that much future production.
I usually build the warrior at the halfway mark on the food storage (as food and production are normally equal) then start building the settler, but I'm not sure when I typically finish him. Sometimes I need to delay settler production if I have too much production though (yeah, I need to watch out for that and build warriors instead).
OK. Suppose your city isn't great and adds +2 food and +2 shields per turn.
You get to size 2 in 10 turns, and you've made a warrior and 10 shields towards a settler. But ideally, you RB'd maybe another 10 shields along the way, so you have 20 now. And at size 2, the city should make about 4 or 5 s/t (two workers on forests, hopefully with no waste), so you finish the settler in about 4 more turns. The doubling-cycle lasts about 14 turns (plus time spent moving settlers to new sites). You can do better with a whale nearby, or after monarchy (+3 and +3 is pretty normal then).
20 food are required to grow a size-1 city to size 2, and 10 shields are required to build a warrior. If the city makes 4 food per turn, it will grow in 5 turns. If it makes 1 shield per turn, it will be occupied in 10 turns. Unless I buy or import the warrior (or adjust the city's resource squares - by taking some food and usually some trade, and maybe exchanging it for production), this city will enter disorder, and then I need to take a resource square anyway, like I should have done earlier, to support an entertainer. If this is one of my initial cities, buying is probably not possible (unless I found "valuable metal deposits" earlier), and importing might not be practical. If I adjust my squares, it's usually to take workers from jungle fruit, impeding taxes, research, and growth, and sometimes I can put them onto ordinary forest to even food and production at 3.
OK, I see what you meant. But the "+4f, +1s" is fairly unusual. Can't you at least get +2 s/t by placing a worker on shielded grass, or even place a worker on forest for 1-2 turns, to make the warrior on time? I realize this will delay your growth a bit, but with +4f, that's no problem.
Is there any way to save a history of a game (without saving each turn in a unique SAV file)? I need to show my attempts and ask what I should have done.
No, I don't think there's any automated way to do it. But many players keep detailed "logs". Also, you might try posting a 2500BC save here. We might be able to give you some advice just from that.
"Incremental". For example, if you have 5 shields and buy a settler in one step, you must pay a lot (approx 120 gold I think). That's "RB". But if you buy a warrior and switch production to a settler (before the warrior is built, of course) you can then buy the settler for less. Or you can repeat the trick with a phalanx (if you have bronze working) or wait a few turns to buy. Many good options, called "IRB".

I figured an archer should explore for more huts, like a mounted unit.
I'm not sure all strong players agree with me on this, but I despise Archers, because they are so slow. I'd rather have the +15 shields towards my next settler. Same for legions. But these units are nice to have on defense, if you expect trouble.
Ooh, that sounds hard to sync.
Yep, it takes some practice. But it's pretty easy if both cities are on the same river, or a short road. Or if you can do it with several nearby citiies. Each city should be size 1 approx 70% of the time (see above) so your odds are pretty good.
I'm on a large map, 5 billion years, and anytime I don't find an enemy something else crippling happens (I miss Colossus and/or Hanging Gardens, my 13th size-1 city revolts, I miss the 5-city mark by about 500 years,...).
Now, we are talking about a slightly later phase of the game. You should have out-grown the AI by now, and should normally be able to make most any Wonder you want before the AI. You need to go for Trade ASAP after monarchy, so you can make vans, to make WoWs. I usually devote a cluster of 4 cities near my capital to make those vans, while the other cities continue with ICS growth. At Deity, you'll need HG soon, so you'll also need pottery. If HG is not urgent, I'll make Marco first, to get techs like pottery from the AI. Colossus isn't too important for EC, but you probably should be able to build it if you want.
Of course, things don't always go right, and you will have some revolts, and some WoWs lost to the AI, etc. In the game I mentioned earlier, I had 20 cities before I could reach Trade, and build HG, so I had LOTS of riots. But I was still glad to have those cities.
BTW, what diplomatic attitude should I take? So far, I've been using appeasement. I became a pet civ once; I got a strategic position with an equal once (when my 13th city was born mad); I made peace that was immediately broken by them at least once; I think I allied with someone who quickly wanted my blood once...
Hmmm. In my games, I rarely have much unit-to-unit contact with the AI until I have about 10-20 cities. By then, I am much bigger than them (do you play mostly on small maps, perhaps?). So, they usually want peace and I am not afraid of war. When another civ demands tribute, even at a bad moment, I usually say no, and they usually back off. I'll agree to peace if they offer it and I have obvious weaknesses, like empty cities. I never make alliances. Sometimes they attack and I lose, but not very often.
Fortify a warrior on a hill? An elephant would make quick work of him, and I think anything with an attack higher than 1 would have a fair chance to beat him. How many warriors (per city) should be sent out, anyway (excluding the ones from size-1 cities)?
You are talking about problems that I [almost] never have. Usually, by the time an AI has polytheism, I am invading them, not defending my own lands. In the odd event that an AI elephant came near, I would try to defend with an elephant, or a horseman or a diplomat (but the fortified warrior on the hill would probably put the ele into yellow or red at least). Hopefully, I'd have a good river/road system by then and at least one of these units built already (see my previous post).
I meant a total of 1 warrior, or 2 at the most, to stop a wandering AI horseman [the warrior will usually win], or delay a slightly stronger AI unit. If you have a few turns to prepare, you should be able to defeat most any single AI unit by attacking first. If you are defending against many AI units at once, something has already gone wrong with your game.
I am much more worried about lightning-fast attacks by sea on my exposed port cities (which I often leave undefended, especially on lower playing levels). I've lost a couple of GOTMs that way, but never by a land elephant.
The one time I went beyond 15 (the same game where my 13th was born mad), I initially got a pirate in about my 17th or 18th. Disorder was erupting all over (including near the capital); a city typically required as many units as it had citizens for martial law, and a size-5 (maybe 4, too) could only be calmed by a temple.
Ugggh. Sounds like you didn't build HG ? For ICS, you normally don't want size 5 cities [unless you have a strong reason, like trade]. And you don't want temples, because HG is so much more efficient. It should become a top priority as you near the 10-city mark. Mike's is probaby even more efficient than HG, but usually you can't wait for monotheism.
What constitutes "ready for conquest"? Also:Like what? What is "the big picture," particularly before 400 B.C.? What kinds of goals? I normally start building whatever is needed when I see the need, wait for it to be produced, then use it, so I don't normally formulate goals (but then I'm accustomed to late conquest in Prince).
At some point in the game, you have enough cities to make the boats, crusaders, and WoWs you expect to need, and it's time to attack. When I was first learning EC, I needed about 100 crusaders to win, so I needed lots of cities (much less now). Now, I feel "ready for conquest" when I have mapmaking and polytheism, about 15-20 cities, and a few key WoWs (such as MPE, LH, HG). There's not much point to building barracks before that, but later you might want 5-10 of them. You can even omit them, but I think they are usually worth the cost at that point.
We talked about the goal of building HG before you make TOO many cities, and that requires some planning [Trade, Pottery, city-clusters and roads for your vans help]. After MPE, I am thinking a lot about how long the conquest will take and how to get my units to the distant AI capitals on time.
And there are lots of options in an EC game. You may aim for a big fat civ (100+ cities) or a small lean one (maybe 15 cities). You may aim for a big economy (vans from your bigger cities to overseas AI ports, shipchains of 40 boats, etc) or you can omit all that. Your decisions may be influenced by the map size, playing level, or just your mood.
You jumped from Prince to Deity ?

Brave! But wouldn't you enjoy moving up more gradually - mastering one level at a time?