We have analyzed both high-resolution optical absorption-line spectra and UV spectra obtained with IUE of the LMC SN 1987A, in order to determine abundances and physical conditions in the various neutral interstellar clouds along the line of sight to the supernova (SN). We have used a flat-fielding procedure to enhance the signal-to-noise ratios (S/Ns) and the reliability of weak features in the UV spectra and have modeled the UV line profiles using the component structure derived from the higher resolution, high-S/N optical spectra of Ca II and Na I. Fits to the Ca II, Ca I, and Na I absorption-line profiles reveal (at least) 46 components, at velocities -24 km s-1v296 km s-1, which can be associated with the 10 component groups discernible in the lower resolution UV spectra. From the UV spectra, we determined component-group column densities for C I, Mg I, Mg II, Al II, Si II, P II, Cl I, Ti?II, Cr II, Mn II, Fe II, Ni II, and Zn II?with 1 ? uncertainties less than 0.1 dex in many cases. These are the most extensive and accurate abundances yet measured for the neutral ISM in the LMC. The component velocities, the patterns of relative elemental abundances [X/Zn] and [X/Fe], and various diagnostic ratios have been used to estimate the locations and physical characteristics [N(H), T, n] of these component groups. (Systematic differences among the diagnostic ratios make the derived physical properties somewhat uncertain, however.) The components at low velocities (5 km s-1v23 km s-1) have relative abundances and values for the diagnostic ratios very similar to those found for warm, diffuse Galactic disk clouds and likely are due to a mixture of warm and cool gas in the Galactic disk. The components at velocities 56 km s-1v90 km s-1 are due to a mixture of warm and cool gas, apparently with negligible depletions, in the Galactic halo. The two intermediate-velocity component groups (109 km s-1v140 km s-1 and 155 km s-1v176 km s-1) both have relative abundances similar to those found for Galactic halo clouds. These warm (T4500 K), partially ionized clouds are probably located in the Galactic halo and in the LMC, respectively. The components at velocities 191 km s-1v225 km s-1 also have relative abundances similar to those in the halo clouds but are likely due to gas in the LMC, perhaps very close to the SN. The component groups at 238 km s-1v255 km s-1 and 265 km s-1v270 km s-1 are probably located on opposite sides of the main LMC component group (at velocities 275 km s-1v296 km s-1) (using absorption-line data for several other adjacent lines of sight and the structure inferred from SN light-echo observations). Although the relative abundances and diagnostic ratios for those three LMC groups are similar to those found for warm, low-density Galactic disk clouds, the widths of individual components seen in very high resolution spectra of Na I and K I imply that T is generally less than about 1500 K. Higher N(Na I)/N(Ca II) ratios, the presence of CH, and the C I fine structure level populations suggest that the main LMC group contains both cool and warm gas. For the LMC components, the total N(H) estimated from the observed relative abundances and inferred depletions is consistent with the value obtained from Ly? absorption toward the neighboring star Sk -69?203, after accounting for differences in reddening and for an overall subsolar metallicity of 0.2-0.3 dex for the LMC ISM. Since the relative abundance patterns determined for stars and gaseous nebulae in both the SMC and the LMC appear to be similar to the solar pattern (for the elements whose interstellar abundances we have considered), the similarities in relative gas-phase interstellar abundances in our Galaxy and in the Magellanic Clouds suggest that the dust depletions follow similar patterns as well?despite differences in metallicity and dust-to-gas ratio among the three galaxies. These local relative abundance/depletion patterns may thus be used to infer total (gas+dust) abundances for QSO absorption-line systems at various redshifts.