  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]
