Force = Mass ⋅ Acceleration I have a doubt on the formula,
$$F=m\cdot a$$ 
In pushing the wall, as per formula,  $F= m\cdot 0=0$ as there is no acceleration hence zero. Therefore force become zero. But we have applied some force which is non zero.
How can then the force be zero?
 A: The total force on the wall includes the part provided by your hand, and also the part provided by the floor and other things to which the wall is attached. This total force on the wall all adds up to zero.
Similarly, the total force on yourself includes the part on your hand and also the friction between your feet and the floor. All this adds to zero in total (if you are not accelerating).
A: Here F does not mean the force that you apply, but it is the sum of all the forces that are being applied. The wall is also applying an equal opposite force F back. This force is applied by the rigidity of the wall and by the its contact with the rest of the room.
$$ F_{ext}=M*a_{CM}$$
And even more fundamental is $$ F_{ext}=\frac{dp}{dt}$$ where P is the momentum.
A: The correct formula is $\sum \vec F = m\vec a$. FOr using this formula first you need to define a system. In your case the system is you and the wall. And as net force is zero so acceleration becomes zero.
The net force is zero because, in reaction to the force that you applied on the wall, the wall also applied the same force but in the opposite direction. 
A: Now starting from scratch, what we know from Newton's 2nd law is given below:-
$$\vec{F}_{net}=m\vec{a}_{net}$$
When we push the wall the forces acting on it are:- 1) Friction in opposite direction of your push. 2)Your pushing force. 3)Normal reaction from earth. 4)Weight. 5)Downward normal reaction from ceiling. As,
$$\vec{a}_{net}=0$$
$$\vec{f} +\vec{N}_{e}+m\vec{g} +\vec{N}_{c}+\vec{F}_{you}=0$$
I think these two equations in itself describes the whole mechanics of the problem.
Hope this helps!
