  We report on the magnetic and the electronic properties of the prototype
dilute magnetic semiconductor Ga$_{1-x}$Mn$_x$As using infrared (IR)
spectroscopy. Trends in the ferromagnetic transition temperature $T_C$ with
respect to the IR spectral weight are examined using a sum-rule analysis of IR
conductivity spectra. We find non-monotonic behavior of trends in $T_C$ with
the spectral weight to effective Mn ratio, which suggest a strong
double-exchange component to the FM mechanism, and highlights the important
role of impurity states and localization at the Fermi level. Spectroscopic
features of the IR conductivity are tracked as they evolve with temperature,
doping, annealing, As-antisite compensation, and are found only to be
consistent with an Mn-induced IB scenario. Furthermore, our detailed
exploration of these spectral features demonstrates that seemingly conflicting
trends reported in the literature regarding a broad mid-IR resonance with
respect to carrier density in Ga$_{1-x}$Mn$_x$As are in fact not contradictory.
Our study thus provides a consistent experimental picture of the magnetic and
electronic properties of Ga$_{1-x}$Mn$_x$As.
