  We present new determinations of the stellar-to-halo mass relation (SHMR) at
$z=0-10$ that match the evolution of the galaxy stellar mass function, the
SFR$-M_*$ relation,and the cosmic star formation rate. We utilize a compilation
of 40 observational studies from the literature and correct them for potential
biases. Using our robust determinations of halo mass assembly and the SHMR, we
infer star formation histories, merger rates, and structural properties for
average galaxies, combining star-forming and quenched galaxies. Our main
findings: (1) The halo mass $M_{50}$ above which 50\% of galaxies are quenched
coincides with sSFR/sMAR$\sim1$, where sMAR is the specific halo mass accretion
rate. (2) $M_{50}$ increases with redshift, presumably due to cold streams
being more efficient at high redshift while virial shocks and AGN feedback
become more relevant at lower redshifts. (3) The ratio sSFR/sMAR has a peak
value, which occurs around $M_{\rm vir}\sim2\times10^{11}M_{\odot}$. (4) The
stellar mass density within 1 kpc, $\Sigma_1$, is a good indicator of the
galactic global sSFR. (5) Galaxies are statistically quenched after they reach
a maximum in $\Sigma_1$, consistent with theoretical expectations of the gas
compaction model; this maximum depends on redshift. (6) In-situ star formation
is responsible for most galactic stellar mass growth, especially for lower-mass
galaxies. (7) Galaxies grow inside out. The marked change in the slope of the
size--mass relation when galaxies became quenched, from $d\log R_{\rm
eff}/d\log M_*\sim0.35$ to $\sim2.5$, could be the result of dry minor mergers.
