This study presents Keck optical and infrared spectroscopy of the rest-frame ultraviolet and optical emission lines in two Ly alpha -emitting galaxies at z > 2. These data provide insight on the evolution of fundamental galaxy scaling relations at early epochs, especially the luminosity-velocity and luminosity-metallicity relations. Spectral diagnostics suggest that the Coup galaxy at z = 2.3 and Lynx Fourre 2-9691, a serendipitously discovered, luminous Lyman drop galaxy at z = 2.9, are star-forming galaxies without active nuclei. Lynx 2-9691 exhibits extended [O III] emission over a diameter of greater than 28 kpc, reminiscent of the Ly alpha nebulae discovered near Lyman drop galaxies. We estimate star formation rates of 59 and 111 M. yr(-1), respectively, from Balmer recombination line luminosities, 2-3 times higher than inferred from the ultraviolet continuum. The ratios of strong nebular emission lines indicate subsolar oxygen abundances in the range 8.2 < 12 + log (O/H) < 8.8 (Z = 0.25-0.95 Z.). Interestingly, Galactic metal-rich globular clusters have similar metallicities, consistent with the idea that we could be seeing the formation of galaxies like the Milky Way at z similar to 3. The measured gas-phase oxygen abundances are greater than 4-10 times higher than the Z < 0.1 Z. metallicities found in damped Ly<alpha> (DLA) absorbers at similar redshifts, indicating that DLA systems trace fundamentally different environments than the vigorously star-forming objects observed here. If this intense star formation activity represents the dominant formation episodes for stars in today's spiral bulges or ellipticals, then the evolved descendants in the local universe should exhibit similarly subsolar metallicities in their dominant stellar populations that formed 8-10 Gyr ago. When these new data are combined with a sample of four other high-redshift spectroscopic results from the literature, we find that star-forming galaxies at z similar to 3 are 2-4 mag more luminous than local spiral galaxies of similar metallicity and thus are offset from the local luminosity-metallicity relation. Their kinematic line widths are sigma (v) = 65-130 km s(-1), making this sample 1-3 mag more luminous than local galaxies of similar line width and mass. Less luminous Lyman drop galaxies need to be studied to see if these deviations are universal or apply to only the most luminous high-redshift galaxies.