With 10 years of high-resolution imaging data now available on the stellar cluster in the Galactic center, we present proper motions for more than 40 stars at projected distances ≤12 from Sagittarius A* (Sgr A*). We find evidence on a ≥2 σ level for radial anisotropy of the cluster of stars within 1'' of Sgr A*. For a brightness limit of K ~ 15.5, we find no evidence for a stationary source at the position of Sgr A* or for a source at this position that would be variable on a timescale of at least several hours to days. On timescales of seconds to tens of minutes, we find no variability at the Sgr A* position on brightness levels K ≤ 13.5. We confirm/find accelerated motion for six stars, with four stars having passed the pericenter of their orbits during the observed time span. We calculated/constrained the orbital parameters of these stars. All orbits have moderate to high eccentricities. We discuss the possible bias in detecting preferentially orbits with high eccentricities and find that measured values of e > 0.9 might be detected by about a factor of 1.5-2 more frequently. We find that the center of acceleration for all the orbits coincides with the radio position of Sgr A*. From the orbit of the star S2, the currently most tightly constrained one, we determine the mass of Sgr A* to be × 10⁶ M and its position to 2.0 ± 2.4 mas east and 2.7 ± 4.5 mas south of the nominal radio position. The mass estimate for the central dark mass from the orbit of S2 is fully consistent with the mass estimate of × 10⁶ M obtained from stellar proper motions within 12 of Sgr A* using a Leonard-Merritt mass estimator. We find that radio astronomical observations of the proper motion of Sgr A*, in combination with its intrinsic source size, place at the moment the tightest constraints on the mass density of Sgr A*, which must exceed ρ > 3 × 10¹⁹ M pc^-3.