  We present Keck/OSIRIS laser guide-star assisted adaptive optics (LGSAO)
integral field spectroscopy of [O III] 5007 Angstrom nebular emission from
twelve galaxies hosting optically faint (R = 20 - 25; nuLnu ~ 10^44 -10^46
erg/s) active galactic nuclei (AGN) at redshift z ~ 2 - 3. In combination with
deep Hubble Space Telescope Wide Field Camera 3 rest-frame optical imaging,
Keck/MOSFIRE rest-optical spectroscopy, and Keck/KCWI rest-UV integral field
spectroscopy we demonstrate that both the continuum and emission-line structure
of these sources exhibit a wide range of morphologies from compact isolated
point sources to double-AGN merging systems with extensive ~ 50 kpc tidal
tails. One of the twelve galaxies previously known to exhibit a proximate
damped LyA system coincident in redshift with the galaxy shows evidence for
both an extended [O III] narrow-line emission region and spatially offset LyA
emission (with morphologically distinct blueshifted and redshifted components)
indicative of large scale gas flows photoionized by the central AGN. We do not
find widespread evidence of star formation in the host galaxies surrounding
these AGN; the [O III] velocity dispersions tend to be high (sigma = 100 - 500
km/s), the continuum morphologies are much more compact than a mass-matched
star forming comparison sample, and the diagnostic nebular emission line ratios
are dominated by an AGN-like ionizing spectrum. The sample is most consistent
with a population of AGN that radiate at approximately their Eddington limit
and photoionize extended [O III] nebulae whose characteristic sizes scale
approximately as the square root of the AGN luminosity.
