That's not correct. It matters a very great deal that the prop is turning.
Consider a small volume of air immediately in front of the propeller blade (small compared to the width of the propeller blade).
The air is moving forward at windspeed, but the cart (and prop) are moving forward faster. So if the prop were not turning, the blade would contact that small volume of air and push it out of the way, transferring momentum from the cart to the air and creating a reaction force which acts to slow the cart.
But the propeller is turning. As the propeller advances towards the small volume of air, the propeller blade turns such that the spot on the blade which is immediately behind the volume of air is further across the face of the blade and, oweing to the pitch of the blade, it is further back. Although the cart has advanced further than the air, the part of the blade behind the volume of air is actually slightly further away. You can see, I hope, that this would tend to produce a lower air pressure in front of the blade and consequently divert the air in front, transferring momentum from the air to the blade and pushing the blade forwards.
If the slower WS air is struck by the faster prop, the transfer of energy takes place over a very, very, short time period. Only when considered as a flow - an aggregate over time - does it make any sense to talk of the blade's motion. However, if that is considered as you suggest, the blade would "move away" from the molecule, so the force would be less than if it did not. It would be self-defeating when hoping to gain energy. Fortunately, rotors do not work like that.
The opposite happens on the rear face of the propeller blade: Each small portion of air, moving forward at windspeed, which comes into contact with the propeller blade finds, that although the propeller as a whole is moving forwards faster than the air, the point in contact with that volume of air is a spot on the blade which is constantly moving across the face of the blade and is advancing more slowly than the wind. So the wind behind the blade transfers momentum to the blade and speeds the cart up.
Same. The acceleration would fall, as the blade moves away. Rather like hitting a ball with a bat, and moving it backwards as you try to hit it the other way....but...
The shape of the airfoil causes the air to follow its profile, such that efficient transfer can take place. That is a
useful by-product of what is known as the no-slip condition,
http://en.wikipedia.org/wiki/No-slip_condition, and rotors are designed to exploit it.
Drag may be seen as a loss, but without it, the rotor efficiency would be very low. That is what so many "blue skies" thinkers and their imaginary props, forget. When losses are wished away, the mechanism that creates what they seek, is wished away too. It is for that reason, that all such analysis are "static", and use only algebra, otherwise the game stops there.
Impact time affects the transfer to or from the blade, so the longer the air stays in contact, the better. Ask a golfer. Keep the head in contact with the ball for as long as it is possible to apply force. Or study impact theory, JBTH.
The same effect accounts for the large static thrust, that falls when the prop is in motion.