Case A: If there is no power at the motor the motor will free spin there is no friction. So in all cases if motor is not connected to generator vehicle will not move in any direction.
Incorrect, the wheel can be locked so it does not spin freely (a worm drive will do that). A wheel that is not turning requires zero power, since power = speed of rotation x torque.
Case B: Vehicle can only move to the right if there is an energy storage device and slip stick hysteresis. In this theoretical analysis there is no specified energy storage device so no movement is possible as net power will always be zero.
You can only have 1W at the motor if you get that from the generator and the two are opposite so will cancel Pnet = Pout - Pin (conservation of energy)
Again, incorrect. If the motor wheel is stationary then it consumes no power, the vehicle is stationary, and 10 W is available to be used from the generator. Take any small amount of power from the generator and supply it to the motor, and the wheel can turn clockwise, moving the vehicle to the right. We have a perfect system with no slipping anywhere.
Case C: There is no way for that vehicle to move to the right. Not quite sure how your intuition works but you have a vehicle with no on board energy source and is on two treadmills both moving from right to left.
This is the point. I don't use intuition, since intuition can give wrong answers. I use engineering analysis instead.
The cart can move to the right if the motor wheel turns faster than 4 m/s. There is ample power available from the generator to power the motor, since the generator wheel is turning at 10 m/s.
If you want a different way of looking at this, the "wind" and the "road" are moving at different speeds (4 m/s and 10 m/s), and the vehicle is in contact with both of them. The vehicle is therefore able to use the difference in speeds (10 − 4 = 6 m/s) as a power source to tap into, and it can use this power source to move in any direction it wishes by turning the motor appropriately.