  We construct the Numerical Galaxy Catalog ($\nu$GC), based on a semi-analytic
model of galaxy formation combined with high-resolution N-body simulations in a
$\Lambda$-dominated flat cold dark matter ($\Lambda$CDM) cosmological model.
The model includes several essential ingredients for galaxy formation, such as
merging histories of dark halos directly taken from N-body simulations,
radiative gas cooling, star formation, heating by supernova explosions
(supernova feedback), mergers of galaxies, population synthesis, and extinction
by internal dust and intervening HI clouds. As the first paper in a series
using this model, we focus on basic photometric, structural and kinematical
properties of galaxies at present and high redshifts. Two sets of model
parameters are examined, strong and weak supernova feedback models, which are
in good agreement with observational luminosity functions of local galaxies in
a range of observational uncertainty. Both models agree well with many
observations such as cold gas mass-to-stellar luminosity ratios of spiral
galaxies, HI mass functions, galaxy sizes, faint galaxy number counts and
photometric redshift distributions in optical pass-bands, isophotal angular
sizes, and cosmic star formation rates. In particular, the strong supernova
feedback model is in much better agreement with near-infrared (K'-band) faint
galaxy number counts and redshift distribution than the weak feedback model and
our previous semi-analytic models based on the extended Press-Schechter
formalism. (Abridged)
