Extremely Metal-Poor Stars. VIII. High-Resolution, High-Signal-To-Noise Analysis of Five Stars with [Fe/H] < -3.5

High-resolution, high-signal-to-noise (<S/N> = 85) spectra have been obtained for five stars -- CD-24:17504, CD-38:245, CS 22172-002, CS 22885-096, and CS 22949-037 -- having [Fe/H] < -3.5 according to previous lower S/N material. LTE model-atmosphere techniques are used to determine [Fe/H] and relative abundances, or their limits, for some 18 elements, and to constrain more tightly the early enrichment history of the Galaxy than is possible based on previous analyses. We compare our results with high-quality higher-abundance literature data for other metal-poor stars and with the canonical Galactic chemical enrichment results of Timmes et al. (1995) and obtain the following basic results: (1) Large supersolar values of [C/Fe] and [N/Fe], not predicted by the canonical models, exist at lowest abundance. For C at least, the result is difficult to attribute to internal mixing effects; (2) We confirm that there is {\it no upward trend} in [$\alpha$/Fe] as a function of [Fe/H], in contradistinction to some reports of the behavior of [O/Fe]; (3) The abundances of aluminum, after correction for non-LTE effects, are in fair accord with theoretical prediction; (4) We confirm earlier results concerning the Fe-peak elements that [Cr/Fe] and [Mn/Fe] decrease at lowest abundance, while [Co/Fe] increases -- behaviors that had not been predicted. We find, however, that [Ni/Fe] does not vary with [Fe/H], and at [Fe/H] ~ -3.7, [Ni/Fe] = +0.08 +/- 0.06. This result appears to be inconsistent with the supernova models of Nakamura et al. (1999) that seek to understand the observed behavior of the Fe-peak elements by varying the position of the model mass cut relative to the Si-burning regions. (abbreviated)