We present a spectroscopic survey of 318 faint (R ~ 27, L ~ 0.1Llow *), Ly alpha -emission-selected galaxies (LAEs) in regions centered on the positions of hyperluminous QSOs (HLQSOs) at 2.5 < z < 3. A sample of 32 LAEs with rest-frame optical emission line spectra from Keck/Multi-Object Spectrometer For InfraRed Exploration (MOSFIRE) are used to interpret the LAE spectra in the context of their systemic redshifts. The fields are part of the Keck Baryonic Structure Survey, which includes substantial ancillary multi-wavelength imaging from both the ground and space. From a quantitative analysis of the diverse Ly alpha spectral morphologies, including line widths, asymmetries, and multi-peaked profiles, we find that peak widths and separations are typically smaller than among samples of more luminous continuum-selected galaxies (Lyman-break galaxies and their analogs; LBGs) at similar redshifts. We find tentative evidence for an association between Ly alpha spectral morphology and external illumination by the nearby HLQSO. Using the MOSFIRE subsample, we find that the peak of the resolved (R approx = 1300) Ly alpha line is shifted by +200 km s super(-1) with respect to systemic across a diverse set of galaxies including both LAEs and LBGs. We also find a small number of objects with significantly blueshifted Ly alpha emission, a potential indicator of accreting gas. The Ly alpha -to-H alpha line ratios measured for the MOSFIRE subset suggest that the LAEs in this sample have Ly alpha escape fractions functionof sub(esc,Ly alpha ) approx = 30%, significantly higher than typical LBG samples. Using redshifts calibrated by our MOSFIRE sample, we construct composite LAE spectra, finding the first evidence for metal-enriched outflows in such intrinsically faint high-redshift galaxies. These outflows have smaller continuum covering fractions (functionof sub(c) approx = 0.3) and velocities (v sub(ave) approx = 100-200 km s super(-1), v sub(max) approx = 500 km s super(-1)) than those associated with typical LBGs, suggesting that the gas covering fraction is a likely driver of the high Ly alpha and Ly-continuum escape fractions of LAEs with respect to LBGs. Our results suggest a similar scaling of outflow velocity with star formation rate (SFR) as is observed at lower redshifts (v sub(outflow) ~ SFR super(0.25)) and indicate that a substantial fraction of gas is ejected with v > v sub(esc). Further observations, including deep spectroscopy in the observed near-IR, will further probe the evolution and enrichment of these galaxies in the context of their gaseous environments.