It may be that the VDV, in Russia's case, is getting an upgraded Sprut-SD on the BMD-4M chassis, with a new FCS, and other goodies. And they may be calling it a "light tank" instead of a "self-propelled anti-tank gun".
As for Ukraine, yes you're right. It's simply a case where they do not need or want a true airborne force, can't field one due to lack of aircraft, and don't know how to use one (just look at the disastrous first assault of Slavyansk, where they used vertical insertions to try to seize objectives, only to be swept off of them by the defendants, and lost 2 helos destroyed 1 damaged).
Well... simply put: they're old. Russian experience in the Chechen and Georgian wars shows that they have many problems, but can still be effective when used properly and with good support. In the case of Ukraine I'd consider the incompetency of the officers and men (some drivers were literally taught at the front how to drive their vehicles), as well as the poor state of the vehicles themselves. A stalled BMP is a much easier target then a moving one.
As for armament, the problem is the optics and FCS (or lack thereof). There is nothing under-powered about the 2A42 auto-cannon. Accuracy is another story.
I'm sorry, my memory isn't the best, were they organic or attached? MBT companies and platoons have been attached to everything from airmobile units, to Ukrainian SpetzNaz units in the course of this conflict. This particular posting means the 79th airmobile will have an organic MBT company in the brigade TO.