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Heavily doped semiconductor thin films are very promising for application in mid-infrared plasmonic devices because the real part of their dielectric function is negative and broadly tunable in the 5 to $50\mu \mathrm{m}$ wavelength range at least. In this work, we investigate the electrodynamics of heavily $n$-type-doped germanium epilayers at infrared frequencies beyond the assumptions of the Drude model. The films are grown on silicon and germanium substrates, are in situ doped with phosphorous in the ${10}^{17}$ to ${10}^{19}$ ${\mathrm{cm}}^{-3}$ range, then screened plasma frequencies in the 100 to $1200{\mathrm{cm}}^{-1}$ range were observed. We employ infrared spectroscopy, pump-probe spectroscopy, and dc transport measurements to determine the tunability of the plasma frequency. Although no plasmonic structures have been realized in this work, we derive estimates of the decay time of mid-infrared plasmons and of their figures of merit for field confinement and for surface plasmon propagation. The average electron scattering rate increases almost linearly with excitation frequency, in agreement with quantum calculations based on a model of the ellipsoidal Fermi surface at the conduction band minimum of germanium accounting for electron scattering with optical phonons and charged impurities. Instead, we found weak dependence of plasmon losses on neutral impurity density. In films where a transient plasma was generated by optical pumping, we found significant dependence of the energy relaxation times in the few-picosecond range on the static doping level of the film, confirming the key but indirect role played by charged impurities in energy relaxation. Our results indicate that underdamped mid-infrared plasma oscillations are attained in $n$-type-doped germanium at room temperature.