We present new results on the physical nature of infrared-luminous sources at 0.5<z<2.8 as revealed by HST/NICMOS imaging and IRS mid-infrared spectroscopy. Our sample consists of 134 galaxies selected at 24\mum with a flux of S(24\mum) > 0.9 mJy. We find many (~60%) of our sources to possess an important bulge and/or central point source component, most of which reveal additional underlying structures after subtraction of a best-fit sersic (or sersic+PSF) profile. Based on visual inspection of the NIC2 images and their residuals, we estimate that ~80% of all our sources are mergers. We calculate lower and upper limits on the merger fraction to be 62% and 91% respectively. At z < 1.5, we observe objects in early (pre-coalescence) merging stages to be mostly disk and star formation dominated, while we find mergers to be mainly bulge-dominated and AGN-starburst composites during coalescence and then AGN-dominated in late stages. This is analogous to what is observed in local ULIRGs. At z \geq 1.5, we find a dramatic rise in the number of objects in pre-coalescence phases of merging, despite an increase in the preponderance of AGN signatures in their mid-IR spectra and luminosities above 10^12.5 L_sun. We further find the majority of mergers at those redshifts to retain a disk-dominated profile during coalescence. We conclude that, albeit still driven by mergers, these high-z ULIRGs are substantially different in nature from their local counterparts and speculate that this is likely due to their higher gas content. Finally, we observe obscured ({\tau}_{9.7\mum} > 3.36) quasars to live in faint and compact hosts and show that these are likely high-redshift analogs of local dense-core mergers. We find late-stage mergers to show predominantly unobscured AGN spectra, but do not observe other morphological classes to occupy any one specific region in the Spoon diagram. [abridged]