Play a non-random map. The random map generator sort of sucks. These maps are not nearly as fun as the maps shipped with the game or the huge quantity of great maps over on http://aow2.heavengames.com/. Some of the maps over at heavengames are more like RPG adventures with meaningful quests and good jokes.

All the official maps are meant for multi player. If you want good single player maps, some of which even puts the campaign maps to shame, try some of the user made maps.

I've tried downloading a scenario (couldn't find just maps) and it still locked my choice to whatever the scenario maker designed. What I want to do is to perhaps click on the portrait and choose my opponent(s) from a list of available characters. Is there a way I can do that?

You can choose the allowed races in 'advanced settings' in the 'generate scenario' tab. Tick the 'Customize Wizard' box in the next screen before you click 'Start!'; then you'll be able to choose which wizard and race you want to play.

So I take it that there is no way to directly choose opponents in SW, just ways to influence which ones are randomly chosen?