  Precise and accurate parameters for late-type (late K and M) dwarf stars are
important for characterization of any orbiting planets, but such determinations
have been hampered by these stars' complex spectra and dissimilarity to the
Sun. We exploit an empirically calibrated method to estimate spectroscopic
effective temperature ($T_{\rm{eff}}$) and the Stefan-Boltzmann law to
determine radii of 183 nearby K7-M7 single stars with a precision of 2-5%. Our
improved stellar parameters enable us to develop model-independent relations
between $T_{\rm{eff}}$ or absolute magnitude and radius, as well as between
color and $T_{\rm{eff}}$. The derived $T_{\rm{eff}}$-radius relation depends
strongly on [Fe/H], as predicted by theory. The relation between absolute $K_S$
magnitude and radius can predict radii accurate to $\simeq$3%. We derive
bolometric corrections to the $VR_CI_CgrizJHK_S$ and Gaia passbands as a
function of color, accurate to 1-3%. We confront the reliability of predictions
from Dartmouth stellar evolution models using a Markov Chain Monte Carlo to
find the values of unobservable model parameters (mass, age) that best
reproduce the observed effective temperature and bolometric flux while
satisfying constraints on distance and metallicity as Bayesian priors. With the
inferred masses we derive a semi-empirical mass-absolute magnitude relation
with a scatter of 2% in mass. The best-agreement models over-predict stellar
$T_{\rm{eff}}$s by an average of 2.2% and under-predict stellar radii by 4.6%,
similar to differences with values from low-mass eclipsing binaries. These
differences are not correlated with metallicity, mass, or indicators of
activity, suggesting issues with the underlying model assumptions e.g.,
opacities or convective mixing length.
