(1) What is the fat cross?
The "fat cross" is the total workable squares for any city. Basically, if you look at your starting city once its borders have expanded once, that's the "fat cross" for the city. Only the squares inside that are usable for that city.
(2) Should I Build cottages on a stream or leave those spots for farms and watermills (later). I usually build cottages in spots that can't build farms.
Like most answers, this one is "it depends". Building cottages on rivers can be quite valuable, as they get an extra commerce (and if you are a financial civ, they'll start at 3 total commerce). This can be invaluable in the early game. On the other hand, it's pretty hard to build watermills somewhere other than a river. And you want the irrigation from fresh water, and rivers are often the best way to get that.
My personal opinion is that it's important to get enough food for your city, so if you need to use some river squares to build farms, do so. Beyond that, figure out what the focus of your city is and adapt from there. If you are going for research/money, then you'll want to focus on cottages. If you're going for production or GPs (Great Persons), then you'll want watermills/farms.
(3) When building watermills, sometimes, it adds value to the city and sometimes not. I cannot figure out WHAT determines that. My thought is that If I knew, I'd reserve those spots for future watermills and build the farms a tile or two away.
Not sure what you mean by "sometimes" adding value. They always add -some- value. But they get better as you develop technologies that improve them, so that might be what you are noticing.
(4) When settling, which tiles are better to build on? For example, if there is a tile with 2 hammers, would you want to build ON it or next to it? I did not pay attention one time and built a city on a wheat field. Wondered the whole game if it really made any difference.
In general, other than squares with resources, the only square that will give your city more starting resources is a hills/plain square (it gives an extra hammer). Beyond that, your city's resources will always be identical. Therefore, it can often be a good idea to settle on the worst square (tundra/desert/etc). But it's generally more important to look at the resources in the area, and try and maximize them.
As for building -on- a resource, most of the time that's a poor idea. For someone learning the game, I'd go so far as to say it's always a bad idea (once you get better at the game, you'll notice the times when it's not). You'll never get the same bonuses for settling on a resources as you would from developing it and working it.
(5) on the barbarians, early on, I can usually kill the wandering ones. The current game, The slipped in to one of my cities and took it over. It was HECK getting it back. Had to throw 3x the archers at it. Lost a lot, but they lost one per round, so eventually (on the 5th or 6th round of the fight) I had more points than they did and got my city back.
Not much of a question here - are you asking how to deal with barbs? The two most common methods are "fog busting" - barbarians can only spawn on squares that are shaded (ie, you can't see them). So the more squares you can see, the less squares the barbs have to spawn on. Generally this involves placing them on hills around your cities, because you can see farther on hills (assuming they aren't surrounded by mountains/forests/jungles).
The other method is roving defense. Especially if you are playing Warlords, getting a few Chariots up will help immensely - Chariots are stronger than any of the early barb units (Axemen have a better base strength, but Chariots get a bonus attacking them). And because they have 2 movement, you can more easily get them to the points of attack.
(6) gonna have to try slavery next game. I don't usually mess with the cities too much.
Slavery can be rather tricky to master. It's easy to see what it does, but it's hard to figure out just when is best to use it. But the best way to do that is by experimentation.
Good luck!
Bh
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