citizens and king
This might be pointless but there two things that make me wonder since we arrived on this mysterious planet.
#1
I have been in haven for months and I don't see one person from our kind who isn't a knight.
#2
Do we have a king? Since we are knights who are to protect and serve our great leader. Is hiding in the shadows like King Tinkinzar?

I sorta thought we were the vanguard wave to set up the planet for further recolonization. Alpha Squad was the first to go in, and we're being sent down to both back up and clean up.
...which is a sad situation, since we all died a long, long time ago.
(I'm gonna make my background story canon, daggumit)

Kozma is just in charge of the gear highest rank I've seen so far is Captain Ozlo.

Our great leader is captain Ozlo - Greta at the Rescue Camp and Valeyn at the Core mentioned him. Kozma is just a quartermaster.

Ozlo must run Spiral HQ technicaly as king, but isn't titled that. We don't know where he is. If SK had a base called Spiral HQ accesable (the stone block to the left of the arcade) we might see him.

I am wondering--
could Barrus be the leader of the Spiral Order? I think Ozlo is just in charge of the Cradle missions.
And BTW the Knights probably have lost their home world.

I find it suspicious that an order of elite knights would be coming out of the rescue camp armed with shovels. I suspect from the number of scientists and technicians we've seen that the spiral order encompasses more than knights.
The frequent references to "recruits" after the crash also make it sound like the original mission of the skylark didn't call for as much of a military presence as we ended up with. I'd guess that the leaders of the spiral order are still stationed on another world, wondering why their peaceful scientific mission hasn't reported in.

I only do King Krogmo's bidding.... Whut? I ain't weird....
~The Mighty Cheese Knight; WeeGee

and I would do it more often if those pesky knights would only stop giving these missions to go on.
Long live lord Krogmo!

While we're discussing the background, I find it odd that we're apparently playing through the relatively early days of being on Cradle storyline-wise, yet some of the gear descriptions suggest that the Spiral Order has been stuck on Cradle for a very long time. Anyone got an ideas about that?

I don't think we've been on the Cradle that long. Though some of the weapon entries and data points mention Spiral vs. Cradle tech, it wouldn't be that far-fetched for a warrior vanguard to quickly engineer (or reverse-engineer) pieces of tech specifically adapted to their situation, a.k.a. our current armaments.
While we're bumbling through the lore, I caught a reference to a Spiral Knights home planet in one of the conversations with Spiral H.Q. - can't remember if it was with Barrus, Kora, or Ferron. Anyone remember what the planet was, and which conversation contains the reference?

Can't say I;ve seen that particular reference. But just going back to what I mentioned, here's the description for an Aegis (emphasis mine):
"A sublime shield entrusted to only the most worthy members of the Spiral Order. Whether its crest is a relic from the Order of old or a creation from the Knight's time on Cradle is not immediately clear. "
To me, that suggests the Order has been here long enough that nobody actually remembers, but if we're playing through the early days, then every knight should know.

nah, 'Order of Old' could mean some time before our ship crashed. Then we dont know if said shield is just a new replica or some shield that survived the crash and was restored.
I believe Kozma is the leader. And for #1, the ship we were in that crashed could have been a military one, that's why we are all knights.
Edit: just remember we crashed on an alien planet, we are just a part of the spiral community.