To clarify on the military, it's not that I was trying to command units directly. I didn't figure out how to interface with the military system at all. The game documentation implies that you should be able to instigate conquest, but I wasn't able to.
Oh, okay, that's different then.
It's done in three steps, from the voting screen. You could just start up a tiny map and grab a country with a single city, so you control the full vote, just to quickly get to grips with it, all steps are "propose vote" to change the desired thing, then when the vote pops up, voting for the desired option.
Step 1: Get your nation's military posture to "offensive", this is required for declaring war.
Step 2: Set an offensive target. You can pick any of your neighbouring kingdoms. The popup should give you some rough idea of military strengths. I'll admit that bit needs a bit more UI work.
Step 3: You can now propose "declare war". Do so, and if it succeeds you'll start a war to conquer the target kingdom.