Energy loss caused by a negative work Let's supposed a box is sliding on a friction-less surface, so it has some KE at this point, and let's imagine an external force is being applied on the opposite direction of the displacement of the box, the work of the external force is negative, so it  will start reducing the KE of the box, and eventually it will stop it... The question here: since energy can neither be destroyed nor created, Where has the initial KE gone? or is it that the negative work can cancel energy? or is it that the KE was converted to another type of energy?
 A: Here kinetic energy is converted/transfered. The entity that does the negative work receives the Kinetic energy of the box. However we cannot determine in which form it receives the energy. In case there is friction some of the kinetic energy will be converted to thermal energy.
A: You have misinterpreted conservation of energy. The statement is that in an ISOLATED system(i.e. a system with no external force), the kinetic energy is conserved$^*$. Clearly, the system you have specified is not isolated. 
$^*$(We can make this statement more general in case we DO HAVE external forces, but they happen to be conservative. In that case, we can define a POTENTIAL energy so that even though the external force acts on the system, the sum of kinetic and potential energy remains constant(conserved).)
Anyway, both of these statements can be put into a single statement, the so called work energy theorem-$$W=\Delta \ KE;$$ i.e. the work done on a system reflects as the change in its kinetic energy. So there is nothing wrong with your system's energy changing here-work is being done upon it. 
If you now consider the entire 'universe'($\equiv$ the system+surroundings), note that as the surroundings exert a force on your system, the energy lost by your system is simply being transferred into the surroundings, so the NET energy of the universe is still constant. If for example the force was exerted by another box, then the reaction force would work on it and cause, in a sense, equal and opposite amount of work, so the net energy change was still zero.
It is essential to specify what your system is. For example, to analyse a ball falling on earth, you could choose your system to be just 'ball' or 'ball+earth'. The analysis would be slightly different in each case, because in the 'ball' system you do have an external force of gravity, while in the earth+ball system, the forces are all internal to the system. Look at Halliday Resnick for a detailed chapter.
A: The idea that the applied force receives kinetic energy from the box is nonsensical.
To understand why it is so, you can pull some moving thing uphill such as a trolley which is sliding dowward a slope. Here the velocity of the troley is downward and opposite to the velocity of your upward-pulling force because your force is
$$F_{\text{applied}} = ma = m \frac{ v^2-v_0^2}{2d} $$
So your velocity reduces the velocity of the moving object, while the velocity of the downward-moving object also reduces your upward-pulling force or upward velocity which means that both you and the object lose KE  because $K =\frac{mv^2}{2}$.
So where the lost KE of the objet and yours go?
The lost KE of the object goes to friction between its wheels and the road surface, which can produce heat or thermal energy $E_{th}$ as being dragged backward and opposite to its downhill motion, and yours goes to the increasing temperature of your physical structure or body as your muscles become more active than they are when you not pulling anything.
If your force is smaller the gravitational force $F_g$ acting on the object, then you will be pulled downward with the downhill-moving object, while you are still pulling the object upward. If your force is equal to the gravitational force acting on the object, it will stop moving and become motionless. If your force is greater than the gravitational force $F_g$ acting on it, it will stop moving downward, and starts moving upward. In the 1st case, you are pulled downward with the moving object, because your force is smaller than the gravitational force acting on it, and the only reason why you are pulled downward with it is that your mass is less than that of the object or your kinetic energy is smaller than that of the object, but not because your kinetic energy is transfered to the object while you are still trying to pull the moving object upward, but not pushing it downward at all.
In the two other cases, the object either stops or reverses its direction, because your force is greater than the gravitational force acting on the object, pulling it downward the slope, or your kinetic energy is greater than that of the object, but not that its kinetic energy is transferred to you while it is still moving downward under the influence of the gravitational force. That is why you feel its weight while pulling it upward. If its kinetic energy were transferred to you as the way how it is so ignorantly understood or fundamentally misunderstood by all the current physicists, you would not feel any weight of the downward-moving object at all. It is just like when you pulling  it downhill, you do not feel its weight at all because it runs downhill by itself under the  pulling force of the gravity and its kinetic energy coming from the gravitational force can be transferred to you if you are attached to it for it to move you downward with it while you do not have to do anything at all.
A: When you apply a force F on the block the block will also exert a reaction force on you and if that force displaces you then you get the block's initial KE.Now you may think of a situation where the block collides against a wall.Here the wall does not move so it shouldn't have any KE.But what we are missing out is that the wall would vibrate under such conditions so the energy would be transferred to the wall in the form of vibrational KE through the reaction force the block exerts on the wall.Some of it may get transferred into sound energy but in the presence of friction it ultimately gets converted to heat and degraded form of energy that can no longer be used.In fact due to increasing entropy more and more energy gets converted into degraded unusable form of energy everyday which ultimately would lead to thermal death of the universe i.e no more energy would be available for use .
