We have undertaken the study of the elemental abundances and kinematic properties of a metal-poor sample of candidate thick-disk stars selected from the Radial Velocity Experiment spectroscopic survey of bright stars to differentiate among the present scenarios of the formation of the thick disk. In this paper, we report on a sample of 214 red giant branch, 31 red clump/horizontal branch, and 74 main-sequence/sub-giant branch metal-poor stars, which serves to augment our previous sample of only giant stars. We find that the thick disk [α/Fe] ratios are enhanced and have little variation (<0.1 dex), in agreement with our previous study. The augmented sample further allows, for the first time, investigation of the gradients in the metal-poor thick disk. For stars with [Fe/H] < -1.2, the thick disk shows very small gradients, <0.03 ± 0.02 dex kpc–1, in α-enhancement, while we find a +0.01 ± 0.04 dex kpc–1 radial gradient and a –0.09 ± 0.05 dex kpc–1 vertical gradient in iron abundance. In addition, we show that the peak of the distribution of orbital eccentricities for our sample agrees better with models in which the stars that comprise the thick disk were formed primarily in the Galaxy, with direct accretion of stars contributing little. Our results thus disfavor direct accretion of stars from dwarf galaxies into the thick disk as a major contributor to the thick-disk population, but cannot discriminate between alternative models for the thick disk, such as those that invoke high-redshift (gas-rich) mergers, heating of a pre-existing thin stellar disk by a minor merger, or efficient radial migration of stars.