Some Questions -- making up my mind, advice needed

[Archonsod]

You can check your graphics card through dxdiag. Start -> run -> dxdiag. It should be listed under Display. Wouldn't say issues with ATI cards are normal, though I think there's an identified bug with them & Doms 3.

As for the improvements in SP : The AI is much improved. Kind of. When it's good, it's very, very good. When it's bad it's pathetic (Monolith pretender, imprisoned and maximum bad scales. Dominion suicide by turn 3...)
Another improvement that hasn't been talked about much (although it goes for MP too) is that there are in effect more nations, hence a lot more variety. Even with the extensions of nations in later era's - MA Ulm feels a lot different to play or face than EA Ulm for example.
The ability to randomise the AI nations is handy, especially in combination with the random map generator. You can start a game not knowing precisely what you'll be up against, adding some interesting challenges (like suddenly finding out you have AE Ermor to go up against in turn 200...)
Most of the other improvements go for either SP or MP equally - new mechanics, more choice in the kind of game you play and similar.
The only real criticism I could make of the SP game at the moment is the flaky AI, but hopefully that will be sorted out in a patch. As I said, when it works right, it can be quite a challenge. However, it can suffer from bouts of stupidity, such as the already mentioned two turn dominion death.