There is a well-known crash in the game if you are alt-tabbed away from Demigod when it connects to a multiplayer match and starts to load the game. Stardock have NOT acknowledged this bug and are not currently working to fix it, according to their own blog posts.
Beyond that, to fix your first problem:
Either download a Mouse-lock program (google it), but that is annoying. . . OR go to the graphics options of Demigod and set it to play Windowed size. And then maximize the window after you click apply (you don't select a resolution for windowed mode. . . it updates it for you whenever you resize the window!)
This way, Demigod behaves like regular windows. . . it still takes up your whole 22" monitor (it's mazimized), and if you click things on your 19" monitor, it doesn't minimize Demigod. . . it just moves the focus to that other window temporarily.
BEWARE: the game will still crash if it connects to an MP game while you are clicked onto a window on your 19" 2nd monitor in this set up. If the game is almost done connecting all players, click on Demigod until the game starts to load to be safe.
I know so much about this cuz I am in the exact same situation, except that my 2nd monitor is only 17" 