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 Infrared Spectrograph mid-infrared spectroscopy. Our sample consists of 134 galaxies selected at 24um with a flux of S(24um)>0.9mJy. 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 active galactic nucleus (AGN)-starburst composites during coalescence and then AGN dominated in late stages. This is analogous to what is observed in local ULIRGs. Finally, we observe obscured ({tau}_9.7um_>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 possess predominantly unobscured AGN spectra, but do not observe other morphological classes to carry any specific combination of {tau}_9.7um_ and polycyclic aromatic hydrocarbon (PAH) equivalent width.