A sample of 27 low-redshift, mostly cool, ultraluminous infrared galaxies (ULIRGs) has been imaged at 1.6 μm with the Hubble Space Telescope (HST) Near-Infrared Camera and Multi-Object Spectrometer (NICMOS). The majority (67%) of the sample's galaxies are multiple-nucleus galaxies with projected separations of up to 17 kpc, and the rest of the sample (33%) are single-nucleus galaxies, as determined by the NICMOS angular resolution limit. The average observed, integrated (host+nucleus) H magnitude of our HST H sample ULIRGs is -24.3, slightly above that of an L* galaxy (MH = -24.2), and 52% of the sample's galaxies have sub-L* luminosities. The ULIRGs in the HST H sample are not generated as a result of the merging of two luminous (i.e., ≥L*) spiral galaxies. Instead, the interactions and mergers occur in general between two, or in some cases more, less massive sub-L* (0.3-0.5L*) galaxies.\ud \ud Only one out of the 49 nuclei identified in the entire HST H sample has the properties of a bright quasar-like nucleus. On average, the brightest nuclei in the HST H sample galaxies (i.e., cool ULIRGs) are 1.2 mag fainter than warm ULIRGs and low-luminosity Bright Quasar Survey quasars (BQS QSOs) and 2.6 mag fainter than high-luminosity BQS QSOs. Since the progenitor galaxies involved in the merger are sub-L* galaxies, the mass of the central black hole in these ULIRGs would be only about (1-2) × 107 M☉, if the bulge-to-black hole mass ratio of nearby galaxies holds for ULIRGs. The estimated mass of the central black hole is similar to that of nearby Seyfert 2 galaxies but at least 1 order of magnitude lower than the massive black holes thought to be located at the center of high-luminosity QSOs. Massive nuclear starbursts with constant star formation rates of 10-40 M☉ yr-1 could contribute significantly to the nuclear H-band flux and are consistent with the observed nuclear H-band magnitudes of the ULIRGs in the HST H sample. An evolutionary merging scenario is proposed for the generation of the different types of ULIRGs and QSOs on the basis of the masses of the progenitors involved in the merging process. According to this scenario, cool ULIRGs would be the end product of the merging of two or more low-mass (0.3L*-0.5L*) disk galaxies. Warm ULIRGs and low-luminosity QSOs would be generated by a merger involving intermediate-mass (0.5 L*) disk galaxies. Under this scenario, warm ULIRGs could still be the dust-enshrouded phases of UV-bright low-luminosity QSOs, but cool ULIRGs, which are most ULIRGs, would not evolve into QSOs.