Yes and no. They have an expanded field of vision but they don't automatically know where every unit is on the map, only the ones in their immediate area. Naval units for instance can see any enemy units that are within their movement range, not just the 2 tiles that the player can see. I'm not sure what the limit for land units with only 1 movement point is.
Yes, that radar as I like to call it is indeed determined by movement of the units. IIRC, 1
units have a grid of 5x5 while 2
have much bigger. And it's even more bigger once Engineering is enabled for the AI or railroads are disposed under.
The ultimate omniscience I was referring to is something even more subtle that can be seen especially in cases of inclosed cities in mountains. Let imagine two little gap that can fit a unit. The nearest to the enemy AI will normally choose the nearest gap to reach the city, but if you block it, they'll choose the other one, even if it demands to walk the half map. Of course, once you block both gaps, the normal behavior is seen again.
The even more subtle aspect is their pathfinding is determined even without having explored the area, which is obvious cheating, but for just cause.
And that crazy omniscience is really seen in all aspects of SoD or assaulting nary reaching places never explored.
Putting that aside, the radar is used as barb protection or immediate unit retaliation. Nevertheless, the AI has a much bigger omniscience that we would imagine. There is not hiding once the AI met.
Man, I would like to play with someone of your knowlegde, but I know you don't want to deal with others.