Checkers,
If the RPM is the same, then the drag is the same, engine running or not - the thing is that if the engine is running, the RPM will be greater than if the prop is windmilling.
I don't think so. Complex flows, twisted blades, variable alpha - the answer is not quite that simple.
Even with fixed blades, the same speed of rotation may be achieved as a combination of power applied and thrust or drag vectors in the plane of rotation. If we fix the speed of rotation, the blade element alpha is controlled only by the TAS (mostly) and the inflow angle. The result is that at low TAS the blade would stop due to high rotational drag so we have to use lots of power to maintain the fixed RPM. This is the same as the take-off case and produces postive thrust. At high TAS, the alpha is such that the prop tends to overspeed through windmilling (rotational thrust) requiring no engine power to maintain the fixed RPM yet producing significant negative thrust, or "drag" as we normally like to call it.
Thus the "drag" vector (along the aircraft flight path) varies quite considerably at the same RPM.
What I think stays the same is the net energy required at that RPM - it is absorbed from the airflow (TAS) or the engine or both. The energy required when the prop is stationary is quite low. When the prop is windmilling, buckets of energy are absorbed.
We need to be careful using the concepts of drag, particularly when we use it as it applies to normal wing sections. When the prop is stationary, it is just a collection of little wing sections aligned with the airflow and we can properly describe the result as form drag in the flightpath axis. However, when the prop is turning, the whole "drag" game gets messy. The blade sections are subject to both form and induced drag relative to their local airflow, but that is not the same axis as the aircraft flight path. When we resolve various forces from the propellor along the aircraft axis we are actually seeing vector combinations of blade section lift and drag - in the windmilling case, the greatest contributor is blade section lift that just happens to point backwards (in an aircraft sense).
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