Okay, I may need to be held back a grade here, but confirm a suspicion for me...
On the subjects of fat crosses and building improvements...
I saw a screen cap somewhere of a dotmapping example where someone had the arms of their city's fat crosses overlapping (so, same lattitude w/ 3 tiles between them). Am I mistaken, or can a tile not be worked by citizens of two seperate cities making this something to avoid if possible. Only time I could see this not being as much of an issue is if the shared tiles were unworkable (desert for example).
Only one city can use a tile although the tile can be switched between both cities by clicking on it in city view.
In most cases, it's true that you don't want to have a
huge overlap of tiles between your cities as it means that the city will never be able to use a decent number of tiles reducing the value of the city compared to the cost of creating, maintaining and improving the city. But some overlap is not a problem if it serves another goal.
For instance, in an area of 50 tiles, I could never place 3 cities without overlap as I'd need a minimum of 3*21=63 tiles for that. But on the other hand, I could never use all tiles with just 2 cities as 2 cities can use a maximum of 2*21=42 tiles. In this case, I'm pretty sure that I'd rather use 3 cities to be able to use all 50 tiles instead of wasting the 8 tiles. It would still mean an average of 16 to 17 tiles per city. On the map, you'd see something like 13 tiles over overlap between the three cities.
I'd probably first place 2 cities in the 50 tile area to claim the area and later in the game add a third to fully utilise the area and find a good way to split the tiles between the three cities.
It's also good to realise that a city will only reach its full size of 21 tiles pretty late in the game. Before that time, you're wasting a lot of tiles if no city is using those tiles and they're just lying there within the fat cross of one of your cities.
Sometimes when 2 cities have a food resource that they both can use, then you can switch between which city uses it. In that way, you can choose which cities grows quickly which can be useful. It's a micromanagement trick.
Also... other than Roads and Forts, is it safe to assume that the only time you'd want to improve a tile outside one of your city's fat crosses would be if you were building infrastructure for a city you plan to establish in the area. Unless that is the case, doing so just wastes time your worker might better spend elsewhere. (As an example, your cultural boundary has expanded several tiles beyond any of your city's fat cross and you plan to place your next city on the edge of or just beyond your cultural border... it might make sense to have a farm or mine in place before the settler gets there and founds the city).
If a resource is inside your culture, improved with its special resource enabling improvement (or a fort in BTS) and connected to your cities, then you get the special benefits of that resource.