I don't know if I missed it in the rules, does a unit at the headquarters need to pay a deployment cost to be moved to a planet?

No, they move with the warlord when the warlord commits to a planet.

No, they move with the warlord when the warlord commits to a planet.

And unlike the Warlord arrive Exhausted.

And currently there's no way to deploy a unit directly to HQ from hand I believe (outside the planet battle ability). Though always room in the design space for that to change I guess.

It's worth noting that planet's Battle Ability puts the unit into play at the HQ rather than "deploying" it to HQ. "Put into play" and "deploy" are separate, distinct mechanics.

Right. I was thinking there might be a new keyword where the unit could put itself "into play" at the HQ based on some trigger or I dunno, lol.

I don't doubt that there will be. Eventually, we will probably see every permutation on how and where a unit can be gotten into play put into effect.