We present the SINS/zC-SINF AO survey of 35 star-forming galaxies, the largest sample with deep adaptive optics-assisted (AO) near-infrared integral field spectroscopy at z~2. The observations, taken with SINFONI at the Very Large Telescope, resolve the Ha and [NII] line emission and kinematics on scales of ~1.5 kpc. In stellar mass, star formation rate, rest-optical colors and size, the AO sample is representative of its parent seeing-limited sample and probes the massive (M* ~ 2x10^9 - 3x10^11 Msun), actively star-forming (SFR ~ 10-600 Msun/yr) part of the z~2 galaxy population over a wide range in colors ((U-V)_rest ~ 0.15-1.5 mag) and half-light radii (R_e,H ~ 1-8.5 kpc). The sample overlaps largely with the main sequence of star-forming galaxies in the same redshift range to a similar K_AB = 23 magnitude limit; it has ~0.3 dex higher median specific SFR, ~0.1 mag bluer median (U-V)_rest color, and ~10% larger median rest-optical size. We describe the observations, data reduction, and extraction of basic flux and kinematic properties. With typically 3-4 times higher resolution and 4-5 times longer integrations (up to 23hr) than the seeing-limited datasets of the same objects, the AO data reveal much more detail in morphology and kinematics. The now complete AO observations confirm the majority of kinematically-classified disks and the typically elevated disk velocity dispersions previously reported based on subsets of the data. We derive typically flat or slightly negative radial [NII]/Ha gradients, with no significant trend with global galaxy properties, kinematic nature, or the presence of an AGN. Azimuthal variations in [NII]/Ha are seen in several sources and are associated with ionized gas outflows, and possible more metal-poor star-forming clumps or small companions. [Abridged]
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