  One bottleneck for the exploitation of data from the $Kepler$ mission for
stellar astrophysics and exoplanet research has been the lack of precise radii
and evolutionary states for most of the observed stars. We report revised radii
of 177,911 $Kepler$ stars derived by combining parallaxes from $Gaia$ Data
Release 2 with the DR25 $Kepler$ Stellar Properties Catalog. The median radius
precision is $\approx$ 8%, a typical improvement by a factor of 4-5 over
previous estimates for typical $Kepler$ stars. We find that $\approx$ 67%
($\approx$ 120,000) of all $Kepler$ targets are main-sequence stars, $\approx$
21% ($\approx$ 37,000) are subgiants, and $\approx$ 12% ($\approx$ 21,000) are
red giants, demonstrating that subgiant contamination is less severe than some
previous estimates and that Kepler targets are mostly main-sequence stars.
Using the revised stellar radii, we recalculate the radii for 2123 confirmed
and 1922 candidate exoplanets. We confirm the presence of a gap in the radius
distribution of small, close-in planets, but find that the gap is mostly
limited to incident fluxes $>$ 200$F_\oplus$ and its location may be at a
slightly larger radius (closer to $\approx$ 2$R_\oplus$) when compared to
previous results. Further, we find several confirmed exoplanets occupying a
previously-described "hot super-Earth desert" at high irradiance, show the
relation between gas-giant planet radius and incident flux, and establish a
bona-fide sample of eight confirmed planets and 30 planet candidates with
$R_{\mathrm{p}}$ $<$ 2$R_\oplus$ in circumstellar "habitable zones" (incident
fluxes between 0.25--1.50 $F_\oplus$). The results presented here demonstrate
the potential for transformative characterization of stellar and exoplanet
populations using $Gaia$ data.
