In this work, we study the chemical compositions and kinematic properties of six metal-poor stars with [Fe/H]{<}-2.5 in the Galactic halo. From high-resolution (R~110000) spectroscopic observations obtained with the Lick/Automated Planet Finder, we determined individual abundances for up to 23 elements, to quantitatively evaluate our sample. We identify two carbon-enhanced metal-poor stars (J1630+0953 and J2216+0246) without enhancement in neutron-capture elements (CEMP-no stars), while the rest of our sample stars are carbon-intermediate. By comparing the light-element abundances of the CEMP stars with predicted yields from nonrotating zero-metallicity massive-star models, we find that the possible progenitors of J1630+0953 and J2216+0246 could be in the 13-25M_{sun}_ mass range, with explosion energies (0.3-1.8)x10^51^erg. In addition, the detectable abundance ratios of light and heavy elements suggest that our sample stars are likely formed from a well-mixed gas cloud, which is consistent with previous studies. We also present a kinematic analysis, which suggests that most of our program stars likely belong to the inner-halo population, with orbits passing as close as ~2.9kpc from the Galactic center. We discuss the implications of these results on the critical constraints on the origin and evolution of CEMP stars, as well as the nature of the Population III progenitors of the lowest-metallicity stars in our Galaxy.