Can someone give me some information about the logistics and organization of early Mongol armies, in the lifetime of Genghis?
For me the most exciting part about Mongol armies (especially the early Mongol armies) is the crucial lack of information.
The Mongols themselves left us next to nothing about this matter; first of all this information was so obvious for any Mongol that it was ridiculous to write it down - if you are Mongol you definitely know that, if you're not a Mongol... well, you're not supposed to know it.
As for the foreign sources they often fell victim to the Mongol policy of deliberate disinformation.
That makes it even more interesting.
What were the largest forces used?
It is safe to say that the largest forces used were in tens of thousands in one place.
40 000 - 50 000 is the biggest number.
Considering how many horses every Mongol usually had that was the main constraint on the size of the Mongol armies of that period.
Not counting auxiliary forces of infantry on the place. But that's a different story - that is for the prolonged sieges mostly.
How self-sufficient were they?
Extremely self-sufficient. Moving fast from place to place the Mongol horses can feed themselves from the ground. If not enough - looting is the best way of feeding your men and horses; the war feeds the war, you know...
How autonomous were commanders at the tumen level?
It depends.
If a tumen is included into the larger army - his commander is not too autonomous.
But if a tumen is acting on his own, the commander is pretty much autonomous.
During his lifetime Genghis khan definitely controlled the level of autonomy of any tumen commander on any particular campaign.
There was a system of the number of horsetails on the banner of the commander on any particular assignment, which indicated the scope of power, authority and autonomy of the given general.
If my memory doesn't fail me, the biggest number of horsetails on the banner was nine, which meant that the general had the power of Genghis khan himself on site. Meaning (among other things) the power of life and death without any trial on anybody in the army.
From all we know Subutai was once given such a banner, which clearly demonstrated the highest degree of trust and confidence.
Would a purely steppe force (no Chinese auxiliaries) have contained any infantry, or just horse archers?
No doubt - no infantry. At all. Whatsoever.
Just horse archers...?
Well, it is not documented in the sources too well, but some specialists are convinced that besides horse archers, Genghis Knan (and his successors) had specialized heavily armed cavalry detachments, which were used as shock troops - to break through the enemy formations engaging into melee.
The obvious equivalent is the knights, cataphractes etc.
But like any other Mongol, this 'steppe knight' had a bow and could shoot an arrow straight of course.