Any force is said to have done a positive work on an object if that force or any component of it displaces the object in the direction in which it acts. It can be seen that the kinetic energy of the body increases due to the application of force, therefore the velocity of the body increases which implies that the displacement is positive.

At the time of compressing, the work done by the boy on the ball is positive as it causes the ball to squeeze which is in direction of force applied.
But the ball resists to change its shape and always tries to regain its original shape by opposing the force applied, therefore, the work done by the ball on the boy is negative.

The ratio of work done by both the boys is the respective change in gravitational potential energies of the ball. Taking the ground level as the reference, the change in gravitational potential energy is $m\times g\times h$ where m- mass, g- acceleration due to gravity and h- height from the earth surface. The  change for one ball would be $m\times g\times 5$ and for the other it would be $m\times g\times 7$ after starting from a zero gravitational potential energy from the ground.
$\therefore$The ratio of the work done would be equal to

$\frac{m\times g\times 5}{m\times g\times 7}=5:7$
 

Let initial velocity of the child be $v_1$ and final velocity be $v_2$
$v_1=2m{s}^{-1}$
$v_2=3m{s}^{-1}$
Since the child is in motion, he will have a kinetic energy of $\frac{m{v}^2}{2}$, which will be increased when an external force acts on him.
Work done by the external force on the child $=\frac{m{v}_2^2}{2}–\frac{m{v}_1^2}{2}$

$=\frac{20\times (9–4)}{2}$

$=10\times 5=50J$.

The work done to stop the block would be negative as the force applied would be opposite to the direction of motion so as to stop the moving block. Since the body possesses only kinetic energy (due to its motion) and no potential energy (as there is no mention of its position, so we assume it to be at ground level), the work required to stop the block will be equal to its kinetic energy. 

In order to stop a moving block, we have to apply force in the opposite direction. Since the force is applied in the opposite direction, this will be negative work.  Since the body possesses only kinetic energy (due to its motion) and no potential energy (as the height to which it is raised from ground level is not changing), the work required to stop the block will be equal to its kinetic energy. 

Given, the weight of a man, $W=600N$, $g=10m{s}^{-2}$, kinetic energy, $KE=750J$
Let the mass of the man be $m$ and the velocity be $v$.
We know, $W=mg$
          So, $600=m\times 10$
On solving we get, $m=60kg$
As we know, $KE=\frac{1}{2}m{v}^2$
So, $750=\frac{1}{2}\times 60\times v^2$
      or, $v=5m{s}^{-1}$
 

Maximum energy depends on the work done in compressing the spring which is directly proportional to the height attained by the ball after the spring is released. The gravitational potential energy is expressed as "m x g x h” and it would be the maximum for the height of $9m$. So, he has to do maximum work to compress the spring when the ball rises to a height of $9m$.

Mechanical energy is the sum of kinetic energy and potential energy in a system.
The law of conservation of mechanical energy states that the mechanical energy in a system remains constant as long as the forces applied are conservative(doesn't depend on the path).