You have already used the following fact in your statement:

For every representation $\rho: G\to \mathrm{Aut}_R(A)$ ($A$ an $R$-module) we can give $A$ the structure of an $R[G]$-module by $g\cdot a = \rho_g(a)$. Conversely, any torsion free $R[G]$-module $A$ admits a representation $\rho: G \to \mathrm{ Aut}_R(A)$ given by $\rho_g(a) := g\cdot a$.

So studying $R[G]$-modules IS studying representations of $G$. My answer would be if you want to study representations study $R[G]$-modules. $R[G]$-modules aren't like a cure-all, because they don't give you an action to start with. Does anyone know how to classify actions?

1. Studying a fixed $R[G]$-module is the same as studying a fixed representation. So it won't tell you anything about the character tables which involves different representations.
2. You can still use $R[G]$-modules to study representations of infinite groups and galois groups. You can use it to derive properties about galois groups acting on roots of unity or torsion points of elliptic curves. http://en.wikipedia.org/wiki/Kummer_theory
3. The wikipedia article http://en.wikipedia.org/wiki/Group_cohomology tells you that the Ext construction from resolutions is the same as the "standard" construction of group cohomology by cochain complexes.

I'm not sure if my answer was helpful. Hopefully it saved you some time. I don't think I understood what you meant about "taking the fixed elements to get the cohomology".