Try running Dominions with a command line parameter -a. This should set all animated background off. In Windows, make a shortcut and chance the 'Target' line like this:
C:\YourDominionsFolder\Dominions II\Dom2.exe" -u This will remove the animation from the main screen and laboratory. In battles, you can press 'w' to chance the background into black void. If the grid isn't visible, press 'g' to have some impression of the battleground.
Dominions III will most probably just add things but use (mostly?) the same engine, so you can assume that anything that Dominions II works with, also works with the sequel.
Regarding multiplay with only one copy of the game:
When playing on the same computer, any number of human players can be present. Also, if you open the turn sent by your friend on your computer, and then finish without editing anything, those unchanged orders will be counted as if they were done on your computer. It works that way, but doesn't work that well when playing *against* your friend...
I would recommend having one of you play with the demo against other's full version, but the demo is of an old version with a nasty bug on one of the Jotunheim themes.