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 $M_{rm vir}sim2times10^{11}M_{odot}$. (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 $dlog R_{rm eff}/dlog M_*sim0.35$ to $sim2.5$, could be the result of dry minor mergers.
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