We examine the nuclear morphology, kinematics, and stellar populations in nearby S0 galaxy NGC 404 using a combination of adaptive optics assisted near-IR integral-field spectroscopy, optical spectroscopy, and Hubble Space Telescope imaging. These observations enable study of the NGC 404 nucleus at a level of detail possible only in the nearest galaxies. The surface brightness profile suggests the presence of three components: a bulge, a nuclear star cluster (NSC), and a central light excess within the cluster at radii < 3 pc. These components have distinct kinematics with modest rotation seen in the NSC and counter-rotation seen in the central excess. Molecular hydrogen emission traces a disk with rotation nearly orthogonal to that of the stars. The stellar populations of the three components are also distinct, with half of the mass of the NSC having ages of ∼ 1 Gyr (perhaps resulting from a galaxy merger), while the bulge is dominated by much older stars. Dynamical modeling of the stellar kinematics gives a total NSC mass of 1.1 × 107 M ⊙. Dynamical detection of a possible intermediate-mass black hole (BH) is hindered by uncertainties in the central stellar mass profile. Assuming a constant mass-to-light ratio, the stellar dynamical modeling suggests a BH mass of <1 × 105 M ⊙, while the molecular hydrogen gas kinematics are best fitted by a BH with a mass of 4.5+3.5 -2.0 × 105 M ⊙. Unresolved and possibly variable dust emission in the near-infrared and active galactic nucleus-like molecular hydrogen emission-line ratios do suggest the presence of an accreting BH in this nearby LINER galaxy.