You know the one I mean: where you tell Lydia "do something for me" and then look at a chair and she goes and sits in it.

I believe the Favor dialogue tab has the responses for this. (I'd call this the "favor system" except that's horribly overloaded with "favor quests" like chopping wood and that's not what I'm talking about.

So how does it work? How do you enter the follower command state? How do you determine which commands are understood and what the response to them might be?

Is it just a case of writing topic responses, conditioning them based on what's under the cursor and executing topic info fragments to make the response happen?

It's one of those awkward areas where all the likely search terms are used for something else...

It's not in Favor. It's in DialogueFollower quest in Dialogue Views/Player Dialogue, and everything is done with one function essentially - SetDoingFavor. Then conditions to check if they're in doing favor state, I don't quite remember what it is. GetFavorState maybe?

IsInFavorState, I'm guessing.

So how would a modder determine the target with which follower has been asked to interact? I'd like to write some non-standard responses for some requests. Or is it all essentially hard-coded and all we can do is write the dialogue?

I suppose worst case I can always use SKSE to detect what's under the cursor and write my own. I'd rather not though

The follower command feature is hard coded. You can trigger it from dialogue using SetDoingFavor but you can also (usually) just hold the activate key on the follower until the special cursor appears. If you want to do something special you'll either have find out what's under the crosshair with SKSE or get very creative. There's always a small chance that the data you need is available in target or one of the other dialogue condition variables.