Why is there conservation of kinetic energy in elastic collision and not in inelastic collision? Why is there conservation of kinetic energy in elastic collision and not in inelastic collision? What would be the difference that lead to conservation of kinetic energy in elastic collision and not in inelastic collision?
We also know that mechanical energy is not converted to heat, light, sound etc. in elastic collision but converted to heat, light, sound etc. in inelastic collision. Why is mechanical energy converted as total energy is conserved in inelastic collision?
 A: 
What is the difference that leads to
  conservation of kinetic energy in elastic collision ?

The difference is only in the properties of the material of a body. If it is elastic (happy ball) it can deform itself (thus absorbing KE) and then recover the original shape, giving back roughly the same amount of KE, which is considered as temporarily stored in the lattices: this question can be of help to you if you want a deeper insight.

You saw this image here: If a body is not elastic (sad ball) the KE will deform the body and this change is irreversible, the KE will be transformed into heat, sound etc. and will not be available anymore as mechanical energy. In this video you can see the enormous difference between a sad and a happy ball of same mass and momentum. If the concept of impulse is not clearly explained there this answer can be of great help

Why is
  mechanical energy converted as total energy is conserved in inelastic
  collision?

Kinetic energy is transformed into an exactly equal quantity of other forms of energy in inelastic collisions, therefore the total energy of the system does not change: KE is not conserved whereas momentum is, but energy in general is conserved anyway
A: The simple answer is that in an elastic collision (for objects >> in mass than typical molecules) energy moves from kinetic to potential then back to kinetic as long as the "elastic limits" of the materials are not exceeded. In other words, as long as they act like springs.
In non-elastic collision the energy goes mostly from kinetic of the colliding masses to kinetic of the particles that make up the masses and surrounding medium - if any. For example, an increase in heat is an increase in the kinetic energy of the particles that make up the material. Same for sound. On a macro scale these are deformations but on a micro scale they are changes in motions of atoms and molecules. Or electrons, in the case of light. Add it all up and you should get the total energy before and after.
I'm not sure about the meaning of the second question. Hope this helps.
A: In Elastic collisions the interaction forces are conservative.  
We can represent the total Energy of the System as : E = U + K
When the particles are far away from each other (separation > 2R) their potential energies remain constant which I choose to be U. This is true except when the particles are in contact which other. 
After collision the colliding particles return to their original configurations - Their dimensions remaining unchanged. 
At the instant of collision the total kinetic energy changes into the potential energy of the System. The Potential energy actually increases when the separation between the particles < 2R. 
During recoil the potential energy is converted back into the kinetic energies of the particle.  
On the other hand in Inelastic collisions the force is non-conservative in nature.
After collisions the particles do not return to their initial configurations -  
Permanent deformations being produced or due to loss of energy in other forms.
This  accounts for the fact that $K_{1} ≠ K_{2}$ 
A: This is the definition of the elastic and inelastic collisions - whether the kinetic energy of the colliding objects is conserved or not.
Note that the total energy is always conserved, but in inelastic collision a part of it is transferred into the internal energy of the colliding objects (typically heat in case of macroscopic objects or excitation energy in microscopic case, although the strict distinction depends on the context).
See also this post.
