  The Neptune-mass GJ 436b is one of the most-studied transiting exoplanets
with repeated measurements of both its thermal emission and transmission
spectra. We build on previous studies to answer outstanding questions about
this planet, including its potentially high metallicity and tidal heating of
its interior. We present new observations of GJ 436b's thermal emission at 3.6
and 4.5 micron, which reduce uncertainties in estimates of GJ 436b's flux at
those wavelengths and demonstrate consistency between Spitzer observations
spanning more than 7 years. We analyze the Spitzer thermal emission photometry
and Hubble WFC3 transmission spectrum in tandem. We use a powerful dual-pronged
modeling approach, comparing these data to both self-consistent and retrieval
models. We vary the metallicity, intrinsic luminosity from tidal heating,
disequilibrium chemistry, and heat redistribution. We also study the effect of
clouds and photochemical hazes on the spectra, but do not find strong evidence
for either. The self-consistent and retrieval modeling combine to suggest that
GJ 436b has a high atmospheric metallicity, with best fits at or above several
hundred times solar metallicity, tidal heating warming its interior with
best-fit intrinsic effective effective temperatures around 300--350 K, and
disequilibrium chemistry. High metal-enrichments (>600x solar) can only occur
from the accretion of rocky, rather than icy, material. Assuming Tint~300--350
K, we find that Q'~2x10^5--10^6, larger than Neptune's Q', and implying a long
tidal circularization timescale for the planet's orbit. We suggest that
Neptune-mass planets may be a more diverse class than previously imagined, with
metal-enhancements potentially spanning several orders of magnitude, to perhaps
over 1000x solar metallicity. High fidelity observations with instruments like
JWST will be critical for characterizing this diversity.
