In Papers I and II of this series, the existence of two distinct halo populations of stars have been found in the solar neighborhood. Precise relative ages and orbital parameters are determined for 67 halo and 16 thick-disk stars having metallicities in the range -1.4 < [Fe/H] < -0.4 to better understand the context of the two halo populations in the formation and evolution of the Galaxy. Ages are derived by comparing the positions of stars in the logT_{eff}-log(g) diagram with isochrones from the Y^2 models interpolated to the exact [Fe/H] and [alpha/Fe] values of each star. Possible systematic errors in T_{eff} and log(g) are considered and corrected. With space velocities from Paper I as initial conditions, orbital integrations have been carried out using a detailed, observationally constrained Milky Way model including a bar and spiral arms. The `high-alpha halo stars have ages 2-3 Gyr larger than the `low-alpha ones. The orbital parameters show very distinct differences between the `high-alpha and `low-alpha halo stars. The `low-alpha ones have r_{max}s to 30-40 kpc, z_{max}s to approx. 18 kpc, and e_{max}s clumped at values greater than 0.85, while the `high-alpha ones, r_{max}s to about 16 kpc, z_{max}s to 6-8 kpc, and e_{max} more or less uniformly distributed over 0.4-1.0. A dual in situ-plus-accretion formation scenario best explains the existence and characteristics of these two halo populations, but one remaining defect is that this model is not consistent regarding the r_{max}s obtained for the in situ `high-alpha component; the predicted values are too small. It appears that omega Cen may have contributed in a significant way to the existence of the `low-alpha component; recent models, including dynamical friction and tidal stripping, have produced orbital parameters as great as those of the `low-alpha component.
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