Newton's third law and Coulomb's law Coulomb's law states that if we have two charges $q_{1}$ and $q_{2}$, then $q_{1}$ will act on $q_{2}$ with a force $$ \textbf{f}_{12}=\frac{q_{1}q_{2}}{r_{12}^2} { \hat {\textbf {r}}_{12}},$$
and $q_{2}$ will similarly act on $q_{1}$ with a force $\textbf{f}_{21}$ such that
$$\,\textbf{f}_{21}=-\textbf{f}_{12}.$$
Suppose the only things we knew was that the repulsive forces vary like $r^{-2}$, and that they depend on the magnitude of the charges involved. Can we infer from these two observations alone that  $\textbf{f}_{21}=-\textbf{f}_{12}$? Or would we need further experiments to establish this equation?
The collinearity can be deduced from symmetrical considerations. What about the magnitude? 
 A: It is worth  repeating that laws in physics are axioms, there is no  proof or derivation other than that the law is necessary, so that a physical mathematical theory can choose those solutions that will fit existing data and, important, will be predictive in new situations. Laws in effect are a distillate of data.
Coulomb's law defines one of the possible forces, so  that Newton's laws can be used  in order to have classical mechanics solutions and predictability in kinematic  problems involving charges.
A: 
Suppose the only things we knew was that the repulsive forces vary like $r^{-2}$, and that they depend on the magnitude of the charges involved. Can we infer from these two observations alone that  $\textbf{f}_{21}=-\textbf{f}_{12}$?

No. In fact, it is not true in general, for a system of two charged particles, that the force acting on charge 1 and the force acting on charge 2 obey Newton's third law at a particular time, given a particular frame of reference. In general there is radiation, Coulomb's law is false, and momentum is exchanged between the charges and the radiation.
A: It's the other way around. Newton explicitly formulated his theory of gravitation so that it would be consistent with conservation of momentum (Newton's third law). Coulomb did the same 100 years later when he formulated what is now called Coulomb's law. 
Yes, Coulumb's law (As well as Newton's Law of Universal Gravitation) do imply Newton's third law of motion. As you can see, it was an integral part of determining that the forces exerted by the particles are equal and opposite.
Conservation of momentum can be proven using elastic collisions
A: Conservation of momentum requires that the forces be reciprocal. The change in the momentum of charge 1 arising from the force between the charges must create an exactly opposite change in the momentum of charge 2 at all times, so the forces must be equal and opposite.
