1) My understanding is that the Soviets invented the concept of the operational level of war, and that it came about because Russian military thinkers began recognizing that with industrialization, and also just Russia's sheer size, that future wars (for Russia anyway) would involve massive forces with large battlefronts and very long supply lines. And so developed the idea of Soviet Deep Battle theory which entails penetrating deep into the enemy's interior, which as a prerequisite would mean mastery of logistics. Now Russia now is not the Soviet military at its height I would think, but, I am sure this school of thinking, and especially given its success by the Russians against the Germans in WWII, still is with the Russians and thus they are aware of its importance. Which probably means this is something they are working towards building in their military buildup.
I am not too familiar with Soviet or Russian military theory but while the Russians have boosted their defense spending the last few years they have focused on weapon systems and C4I, not so much on logistics. While not a perfect one a good example would be to look at their airlift fleet, which consists a much diminished and often poorly maintained fleet of Soviet-era transports supplemented with a very small number of newer aircraft. The same can be said for almost all of their logistical tail, relying on older equipment of greatly varying quality that likely wont' last them long in a high intensity conflict.
2) When you say "logistical depth," how do you mean? Like do you mean physical locations of supplies laid out by NATO forces to support any kind of conflict or do you just mean the overall logistical capabilities of said forces? (like their numbers of logistics personnel and professionalism, numbers of supply and fuel trucks, etc...?)
The overall logistical capabilities of said forces.
3) You say the U.S. and several of our NATO allies; just wondering who those several are?
Most notably France and the UK. While their forces are much smaller than ours they have regularly demonstrated the capability to competently deploy and sustain forces worldwide. It would be a stretch for them to do it on their own but they aren't amateurs at it. For comparison, the Brits and French have or will have 14 (9) and 15 aerial tankers while the Russians have ~20. Again, not a perfect analogy but a telling one as to the balance of their forces.
Another thing to note is the endemic corruption that permeates Russia. While the effects of corruption on the military has diminished in the past few years it has by no means disappeared, with it effecting large weapon system contracts to everyday things like food and housing. That sort of corruption is close to non-existent in most older NATO countries (foreign weapons deals aside).