I just beat King Mode in every category against the AI.
I think they need to rework the AI for King level. Every fucking 4 -6 turns, every Civ (even Ghandi, wtf), even if they have jack shit and are on the other side of the map, demand that I give them a great person/tech/gold. If I deny, they declare war. It doesn't seem to matter how many troops/cities that I have.
Appeasement doesn't change the overall long-term attitude of the enemy toward you, and I believe Civ4 did that. Regardless if you're in war or not, you get interrupted asking for stuff.
I was able to beat all modes except domination fairly easily...but I felt like I was underpowered, but as soon as I got industrial techs, it's like the AI is too dumb to know how to increase production, and I dominate.
Napolean is a bitch if you border him. He'll manage to get a great leader/general and make his cannons and infantry impossible to overcome until you get tanks and bombers.
As the Germans, I couldn't beat Napolean until people were launching stuff into space. As the Romans, I was able to abuse their Republic bonus on Settlers (only consume 1 population) and cheap roads to build a dozen cities, and I used their 1/2 cost wonder bonus to buffer the culture on the bordering cities.