Is it possible to calculate the force exerted by two unequal charge, separately? According to the Coulomb's law, when two equal like point charges are placed 1m from each other the force between them is 9 * 109. 
But if two unequal like point charges are placed, the bigger charge will not be repelled or move, rather the smaller one will be repelled. This is because both the charges will exert different forces on each other and the bigger charge will exert a bigger force. But the coulomb's law tells us about a common force between the two charges, is it possible to calculate the individual forces exerted by two unequal like point charges?
 A: The two forces, A on B,  and B on A,  are equal and opposite.  They are not unequal.
This fact is codified in Coulomb's Law:  the force is proportional to the product of the two charges, regardless of which force you are talking about.  Absolutely the same.
If one of them had a much larger mass than the other, then the heavy one would have a much smaller acceleration, and hence smaller velocity at all times, and if it is very very heavy it  would appear not to move at all.    This is due to Newton's Second Law $a = F/m$.  But the magnitude of the force on object A is identical to the magnitude of the force on object B.
A: The two charges will feel equal and opposite forces 
$$
F = \frac1{4\pi\epsilon_0}\frac{q_1q_2}{r^2}.
$$
If their masses are different, the accelerations will be different.  (For instance, an electron feels roughly 2000 times the acceleration of a proton in the same field, because its mass is much lighter.) But the forces are balanced. We still get to use Newton's third law.
