That's correct.
Nope. It seems like it should. That's what makes this such a good brainteaser. But there are several ways to think about it. When the cart is moving along at 10 mph in a 10 mph tailwind, the cart feels no wind (this is true). The prop blades do still feel wind, because they aren't simply moving downwind with the cart, their also spinning around (i.e. moving crosswind). But let's come back to that.
For now we're going downwind at wind speed - thus as a passenger we feel no wind. However, the road is still passing beneath our wheels at 10 mph (think treadmill). This is a perfectly good energy source. And if we use this energy source, we can spin the prop to provide the necessary resistance in the tailwind to keep us still while the treadmill belt continues to spin our wheels at 10 mph. If there were no velocity differential between the wind and the treadmill belt, we would never be able to extract enough energy from the road (i.e. treadmill belt) to pull this trick off. But that very differential between road and wind puts our prop in a medium that's easier to push against. So the energy available at the wheels is greater than the energy needed at the prop.
Even so, we have to remember that whether the energy comes from the road or the wind is really only a matter of how we do our analysis. This doesn't really make any difference though, because no matter what frame we do our analysis in the experimental results are the same (i.e. speed of the cart, forces on the wheels, prop, prop-shaft, transmission, etc.).
Energy is just a very different animal. In some ways it's like velocity. When the vehicle is going downwind at 10 mph in a 10 mph tailwind, how fast is it going relative to the road? 10 mph. But how fast is it going relative to the air? 0 mph. How can it be both? Velocity only has meaning relative to the reference frame in which it's measured. Kinetic energy is the same in this respect.