We present the AGN, star-forming, and morphological properties of a sample of 13 MIR-luminous (f(24) > 700uJy) IR-bright/optically-faint galaxies (IRBGs, f(24)/f(R) > 1000). While these z~2 sources were drawn from deep Chandra fields with >200 ks X-ray coverage, only 7 are formally detected in the X-ray and four lack X-ray emission at even the 2 sigma level. Spitzer IRS spectra, however, confirm that all of the sources are AGN-dominated in the mid-IR, although half have detectable PAH emission responsible for ~25% of their mid-infrared flux density. When combined with other samples, this indicates that at least 30-40% of luminous IRBGs have star-formation rates in the ULIRG range (~100-2000 Msun/yr). X-ray hardness ratios and MIR to X-ray luminosity ratios indicate that all members of the sample contain heavily X-ray obscured AGN, 80% of which are candidates to be Compton-thick. Furthermore, the mean X-ray luminosity of the sample, log L(2-10 keV)(ergs/s)=44.6, indicates that these IRBGs are Type 2 QSOs, at least from the X-ray perspective. While those sources most heavily obscured in the X-ray are also those most likely to display strong silicate absorption in the mid-IR, silicate absorption does not always accompany X-ray obscuration. Finally, ~70% of the IRBGs are merger candidates, a rate consistent with that of sub-mm galaxies (SMGs), although SMGs appear to be physically larger than IRBGs. These characteristics are consistent with the proposal that these objects represent a later, AGN-dominated, and more relaxed evolutionary stage following soon after the star-formation-dominated one represented by the SMGs.