Why doesn't the equipartition theorem disallow spinning dust?
The first relevant point is that the equipartition theorem only formally applies in thermodynamic equilibrium, which requires that all temperatures be the same.  Since the spinning dust only happens when there are temperature differences, holding to the analogy with a heat engine mentioned in the source, we should not expect that theorem to be the whole story.  Granted, we often apply that theorem even when there are temperature differences, so the issue boils down to, when is the behavior more or less the same when all temperatures are the same, and when is the behavior fundamentally a heat engine?  I will assume the source is correct that this is heat engine-like behavior for spinning grains.

Once you have a heat engine in operation, the sky is the limit on how much energy you can put in any given mode (as long as you can maintain the temperature difference), because a heat engine does work, and work energy never needs to be equipartitioned, you can partition it any way you like by setting up an appropriate apparatus.  The dust particle is therefore a kind of apparatus for doing work, in the presence of temperature differences, and channeling the work energy into rotation.

How and why that heat engine operates and picks rotation to channel the work is a much more subtle issue.  For example, do we need temperature differences across the dust grain itself, or just between the grain and the surroundings?  If the "hypothetical" Crooke's radiometer (meaning, one that works quite differently from actual Crooke's radiometers and actually spins the opposite direction) is a good analogy, then the spin is from temperature differences across the dust grain itself, which warm the gas in the surroundings of the dust grain, producing gas flows that maintain constant pressure.  Those flows require the gas to receive torques, and conservation of angular momentum requires an opposite torque on the dust grain.  

To get torque like that, it seems that you not only need a temperature difference across the dust grain, but it also has to have a kind of "quadrupole" character-- hot-cold on one side of the grain, cold-hot on the other, producing a "handedness" to the temperature structure in the grain.  So we immediately see that one cannot get spin until one has a large enough system to be able to support the concept of "temperature difference" across the object, and molecules are generally not thought of that way-- though maybe in the case of very large molecules, they could be.  So the question then comes down to, how large does a molecule need to get before it can support a concept of spatially nonuniform temperature? I don't know how big that needs to be, but my guess is, very big indeed-- there are certainly polymers that can get very large, but the molecules talked about in astronomy are generally not polymers.

As for the sign of the rotation, that can be very difficult to figure out!  It seems to depend on how the temperature differences get created in response to absorption of a radiation field (or other interactions with the environment, perhaps thermal contact with warmer gas).  Generally speaking, concave sides warm up more quickly than convex, so cause the gas in the concave part to be warmer than in the convex, so a flow sets up from concave to convex.  That acts like a jet engine that pushes the concave side forward.  That this is subtle can be seen from the fact that the higher pressure gas would seem to push the other way on the dust, but it's like the way a sail on a sailboat works-- follow the deflection of the air and conclude that the boat deflects the opposite way.

If this is all correct (see the Wiki on Crooke's radiometers for background information), then I would say the shape of the dust grain you drew produces spin in the direction that you indicated, but that no astrophysical molecules are large enough to produce the same effect.