We study the spectral energy distributions and evolution of a large sample of optically selected quasars from the Sloan Digital Sky Survey that were observed in 323 Chandra images analyzed by the Chandra Multiwavelength Project. Our highest-confidence matched sample includes 1135 X-ray detected quasars in the redshift range 0.2 < z < 5.4, representing some 36 Msec of effective exposure. We provide catalogs of QSO properties, and describe our novel method of calculating X-ray flux upper limits and effective sky coverage. Spectroscopic redshifts are available for about 1/3 of the detected sample; elsewhere, redshifts are estimated photometrically. We detect 56 QSOs with redshift z > 3, substantially expanding the known sample. We find no evidence for evolution out to z ~ 5 for either the X-ray photon index Gamma or for the ratio of optical/UV to X-ray flux alpha ox. About 10% of detected QSOs show best-fit intrinsic absorbing columns greater than 1022 cm-2, but the fraction might reach ~1/3 if most nondetections are absorbed. We confirm a significant correlation between alpha ox and optical luminosity, but it flattens or disappears for fainter (MB -23) active galactic nucleus (AGN) alone. We report significant hardening of Gamma both toward higher X-ray luminosity, and for relatively X-ray loud quasars. These trends may represent a relative increase in nonthermal X-ray emission, and our findings thereby strengthen analogies between Galactic black hole binaries and AGN. For uniformly selected subsamples of narrow-line Seyfert 1s and narrow absorption line QSOs, we find no evidence for unusual distributions of either alpha ox or Gamma .