The reason why the moon can't be eclipsed when it's halfway between the nodes of the orbit because the moon's orbit is at it's maximum deviation, meaning that it is leaving it's course from the sun's path, which would make the shadow fall way below or above the moon. Because of that, the moon can't eclipsed.
Oh my gosh had the same question
The work done by a force is given by:

where
F is the magnitude of the force
d is the distance covered by the object

is the angle between the direction of the force and the direction of motion
In our problem,

,

and

(because the force is parallel to the direction of motion), so the work done by the crane is
The total mass of the mixture would not change.
Think about it as a mix of marbles. Suppose you have a group of 10 marbles. 5 marbles are red, and 5 marbles are blue. If you separate the marbles into 2 different groups by color, you still have 10 marbles in total, but they are separated and no longer a mixture of colors.