We present high-resolution H41α hydrogen recombination line observations of the 12 (3 pc) region surrounding Sagittarius A* (Sgr A*) at 92 GHz using the Millimeter Array at the Owens Valley Radio Observatory (OVRO) with an angular resolution of 7'' × 3'' and velocity resolution of 13 km s-1. New observations of H31α, H35α, H41α, and H44α lines were obtained using the NRAO 12 m telescope, and their relative line strengths are interpreted in terms of various possible emission mechanisms. These NRAO 12 m measurements are the most extensive recombination line survey of this region to date. These OVRO data also represent the highest angular resolution and the highest sensitivity observations of Sgr A West in hydrogen recombination line and continuum emission at millimeter wavelengths. Observations of HCO+(J = 1 → 0) transition at 89 GHz are obtained simultaneously with a 40% improved angular resolution and 4-15 times improved sensitivity over the previously published results, and the distribution and kinematics of the dense molecular gas in the circumnuclear disk (CND) are mapped and compared with those of the ionized gas. The line brightness ratios of the hydrogen recombination lines are consistent with purely spontaneous emission from Te ~ 7000 K gas with ne ~ 2 × 104 cm-3 near LTE condition. A virial analysis suggests that the most prominent molecular gas clumps in the CND have mean densities on the order of 107 cm-3, sufficient to withstand the tidal shear in the Galactic center region. Therefore, these clumps may survive over several dynamical times, and the CND may be a dynamically stable structure. We estimate a total gas mass of about 3 × 105 M for the CND. Our combined analysis of the new high-resolution H41α and HCO+ images and our kinematic model demonstrates a widely spread physical and dynamical link between the molecular gas in the CND and the ionized gas, including along the eastern rim of the CND where a gap was previously suggested.