  We present a statistical study of 180 dust continuum sources identified in 33
massive cluster fields by the ALMA Lensing Cluster Survey (ALCS) over a total
of 133 arcmin$^{2}$ area, homogeneously observed at 1.2 mm. ALCS enables us to
detect extremely faint mm sources by lensing magnification, including
near-infrared (NIR) dark objects showing no counterparts in existing {\it
Hubble Space Telescope} and {\it Spitzer} images. The dust continuum sources
belong to a blind sample ($N=141$) with S/N $\gtrsim$ 5.0 (a purity of $>$
0.99) or a secondary sample ($N=39$) with S/N= $4.0-5.0$ screened by priors.
With the blind sample, we securely derive 1.2-mm number counts down to $\sim7$
$\mu$Jy, and find that the total integrated 1.2mm flux is 20.7$^{+8.5}_{-6.5}$
Jy deg$^{-2}$, resolving $\simeq$ 80 % of the cosmic infrared background light.
The resolved fraction varies by a factor of $0.6-1.1$ due to the completeness
correction depending on the spatial size of the mm emission. We also derive
infrared (IR) luminosity functions (LFs) at $z=0.6-7.5$ with the $1/V_{\rm
max}$ method, finding the redshift evolution of IR LFs characterized by
positive luminosity and negative density evolution. The total (=UV+IR) cosmic
star-formation rate density (SFRD) at $z>4$ is estimated to be
$161^{+25}_{-21}$ % of the established measurements, which were almost
exclusively based on optical$-$NIR surveys. Although our general understanding
of the cosmic SFRD is unlikely to change beyond a factor of 2, these results
add to the weight of evidence for an additional ($\approx 60$ %) SFRD component
contributed by the faint-mm population, including NIR dark objects.
