Thank you, I have not yet performed any other tests already planned.  
However, the game allows to obtain a Commander by building an improvement, it seems appropriate to be able to use it as soon as it is available.

I already have the Commander, no need to train it.  

If I buy resources at the Galactic Market (held by the Korx), I can use them right away, even if I have no starbase that produces these resources or if I have no improvement that produces them.

It's a game, not reality.  

We just need to know how it should work in the game design.
If it 's planned that the Commander can be promoted without the technology (I think he should) or if it is necessary to have researched the "War College" technology to authorize promotions, this should be indicated for the various promotions in the "Promote Citizen" window for the Commander.

the lack of an up to date manual, and the failure of Stardock to update UI elements (specifically descriptions and tool-tips), is certainly the biggest issue.  I agree on this point.  

However, promotions for a citizen not being available until researching the ability to unlock that citizen type, is a feature.  Not a bug.

Perhaps, in your games, you should try having fun instead of nitpicking.

More details on bugs after promoting the Commander in this thread.

From the other thread:

Ship just after Privateer promotion.
It does not work.
Nothing is updated, neither the info nor the icon of the ship.

Privateer promotion is pretty special. So something like an admiral promotes like this:

<OnEvent>OnUnitGenerated</OnEvent>

      <PerformAction>

        <Action>ChangeShip</Action>

        <StringParam>NavigatorCommanderBlueprint</StringParam>

      </PerformAction>

Blueprint is found in CommanderStaticBlueprintDefs.

But a privateer promotes like this:

<OnEvent>OnUnitGenerated</OnEvent>

      <PerformAction>

        <Action>ChangeShipToBestPirateShip</Action>

      </PerformAction>

On the screenshot, it's turn 151, you have no snuggler colony. Are there any pirates on the map? Are you doing something wonky for generating the commander or promoting?

When promoted, privateer instantly changes model if zoomed in, or icon if zoomed out. The lower left window info updates once you un-select and select it again. 

That being said, privateer seems completely detached from your own research. If you have no militarization tech, apart from War Machine so you can upgrade the commander, privateer you get is a "Battleship M0" (escort, large hull, 5 superior railguns, 2 hull platings, hyper drive, life support). This seems to be base privateer model, or base pirate best ship. You can get this on turn 2 (using console to get resources and finish research, and being exterminator for free commander).

Next design in the line replaces 5 superior railguns with 2 superior railguns, 3 basic railguns and a kinetic accelerator.

I have gotten carriers from the upgrade as well, destroyer-l(missile oriented ship) and so on.

As to when designs update and what ship you get, seems very random? Before age of war, it seems it's always that first design you get, but when do pirates update their designs, how do they research...it's very annoying not knowing what you get or when does it change.

I tested with pirate bases set to none (assumed thats what "Pirates are disabled in the options before the game starts." means). Everything still works the same.

Small update on this. I promoted 2 or 3 commanders to privateers by turn 50-60, worked fine, had no military tech other than War Machine. After that it bugged out like reported, you promote nothing happens. That lasted until about turn 100, at which point I finally researched kinetics and missiles fully, and now I can promote to privateer again.