We present a method to flexibly and self-consistently determine individual galaxies' star formation rates (SFRs) from their host haloes' potential well depths, assembly histories, and redshifts. The method is constrained by galaxies' observed stellar mass functions, SFRs (specific and cosmic), quenched fractions, ultraviolet (UV) luminosity functions, UV-stellar mass relations, IRX-UV relations, auto- and cross-correlation functions (including quenched and star-forming subsamples), and quenching dependence on environment; each observable is reproduced over the full redshift range available, up to 0 < z < 10. Key findings include the following: galaxy assembly correlates strongly with halo assembly; quenching correlates strongly with halo mass; quenched fractions at fixed halo mass decrease with increasing redshift; massive quenched galaxies reside in higher-mass haloes than star-forming galaxies at fixed galaxy mass; star-forming and quenched galaxies' star formation histories at fixed mass differ most at z < 0.5; satellites have large scatter in quenching time-scales after infall, and have modestly higher quenched fractions than central galaxies; Planck cosmologies result in up to 0.3 dex lower stellar - halo mass ratios at early times; and, none the less, stellar mass-halo mass ratios rise at z > 5. Also presented are revised stellar mass - halo mass relations for all, quenched, star-forming, central, and satellite galaxies; the dependence of star formation histories on halo mass, stellar mass, and galaxy SSFR; quenched fractions and quenching time-scale distributions for satellites; and predictions for higher-redshift galaxy correlation functions and weak lensing surface densities. The public data release (DR1) includes the massively parallel (>105 cores) implementation (the UNIVERSEMACHINE), the newly compiled and remeasured observational data, derived galaxy formation constraints, and mock catalogues including lightcones.