When we push on a wall, can we say work is being done on the atomic particles in contact with our hand? Since textbooks say work is act transfer of energy, it led me to think of the following assumption:   
Work is being done on the the atoms in the wall, in contact with the hands, when we push (hard) on the wall.
When we push the wall, energy is being transferred as heat energy (due to friction between hand and wall) which increases the kinetic energy of the atomic particles on the wall, thus increasing the temperature of the area in contact. 
I would appreciate if anyone could help me know whether I'm wrong in my assumption and why so.
 A: Friction is a red herring here. You can do work on an object by pushing on it, even if there is no friction.
When you push on a spring, you do work by compressing the spring. The spring deforms (shortens) and the work that you do to deform the spring is stored as potential energy in the spring. The spring pushes back on you with a force that is proportional to its deformation (Hooke's law) and eventually this force equals the force with which you are pushing on the spring, and the spring stops deforming.
When you push on a fixed solid object such as a wall then you do work by compressing the bonds between the atoms and molecules in the wall. These bonds act like tiny springs and store potential energy. The deformation of the wall is very small, but it is real, and it is this deformation that leads to the wall "pushing back" on you.
Of course, if you push too hard and deform the interatomic bonds too much then something breaks !
A: 
Work is being done on the the atoms in the wall, in contact with the
  hands, when we push (hard) on the wall.

At the macroscopic level the wall is considered a rigid body, and since the wall as a whole is not displaced, no work is done.
On the microscopic level you are doing work pushing on the wall by displacing layers of atoms in the wall (but mostly displaces the atoms of your hand). This in effect is stored as strain energy which, after you remove the force, is responsible for moving the atoms back to their original configuration.

When we push the wall, energy is being transferred as heat energy (due
  to friction between hand and wall) which increases the kinetic energy
  of the atomic particles on the wall, thus increasing the temperature
  of the area in contact.

Heat is energy transfer due solely to temperature difference. Any energy transfer by heat to the wall would be due to the temperature of our hand being greater than the wall. 
An increase in temperature due to friction is actually energy transfer by friction work, not by heat. If you rub your hands together on a cold day to keep warm, you are doing friction work. It is the friction work that increases the temperature of your hands skin. In any event, simply pushing on a wall does not involve kinetic friction unless you rub your hands against the wall. 

In any event, simply pushing on a wall does ..... the wall." Had I
  assumed that there is some frictional force between the hand and wall,
  which leads to increase in temperature of wall compared to initial
  condition, would it then be heat (transfer of thermal energy) or work
  done by friction (increase in the mechanical energy of the atoms in
  the wall in contact) or both?

It would be work transfer, not heat transfer. Consider compressing a gas. When you compress a gas its temperature rises. The temperature rise is not due to heat (energy transfer due to temperature difference) but due to work (force times displacement).
Now once the friction work increases the temperature of the surface of the wall (as well as your hand), it can be followed by heat transfer from the surface of the wall to the interior of the wall because the wall surface temperature is  higher than the interior temperature.

That leads me to another question : doesn't heat increases the
  mechanical energy(KE+PE) of the atoms/molecules too? Which would mean
  that heat is also a type work. Please correct me where I am wrong?

Yes, heat transfer can increase the KE and PE of the atoms/molecules of a substance (though we primarily consider it with respect to KE unless a material undergoes a phase change). But the mechanism is not the same as work. The mechanism is energy transfer due solely to temperature difference. Work is not energy transfer due to temperature difference. It is due to force times displacement. 
That said, heat transfer can result in work and work transfer can result in heat. 
If I expose a gas in a cylinder fitted with a piston to a body at a higher temperature than the gas, there will be heat transfer from that body to the gas increasing its temperature. The increase in temperature of the gas causes it to expand against the piston resulting in the gas doing work on its surroundings. 
Likewise, if I compress the gas the work I do transfers energy to the gas increasing its temperature. Then you can have energy transfer from the gas to lower temperature surroundings by means of heat.
Hope this helps.
