We present the rest-frame optical spectroscopic properties of 60 faint (R AB ∼ 27; L ∼ 0.1L * ) Lyα-selected galaxies (LAEs) at z ≈ 2.56. These LAEs also have rest-UV spectra of their Lyα emission line morphologies, which trace the effects of interstellar and circumgalactic gas on the escape of Lyα photons. We find that the LAEs have diverse rest-optical spectra, but their average spectroscopic properties are broadly consistent with the extreme low-metallicity end of the populations of continuumselected galaxies selected at z ≈ 2 − 3. In particular, the LAEs have extremely high [O III] 4 K), low oxygen abundances (12 + log(O/H) ≈ 8.04, Z neb ≈ 0.22Z ), and high excitations with respect to their more luminous continuum-selected analogs. Several of our faintest LAEs have line ratios consistent with even lower metallicities, including six with 12 + log(O/H) ≈ 6.9−7.4 (Z neb ≈ 0.02 − 0.05Z ). We interpret these observations in light of new models of stellar evolution (including binary interactions) that have been shown to produce long-lived populations of hot, massive stars at low metallicities. We find that strong, hard ionizing continua are required to reproduce our observed line ratios, suggesting that faint galaxies are efficient producers of ionizing photons and important analogs of reionizationera galaxies. Furthermore, we investigate the physical trends accompanying Lyα emission across the largest current sample of combined Lyα and rest-optical galaxy spectroscopy, including both the 60 KBSS-Lyα LAEs and 368 more luminous galaxies at similar redshifts. We find that the net Lyα emissivity (parameterized by the Lyα equivalent width) is strongly correlated with nebular excitation and ionization properties and weakly correlated with dust attenuation, suggesting that metallicity plays a strong role in determining the observed properties of these galaxies by modulating their stellar spectra, nebular excitation, and dust content.