We present a statistical study of 180 dust continuum sources identified in 33 massive cluster fields by the Atacama Large Millimeter/submillimeter Array Lensing Cluster Survey (ALCS) over a total of 133arcmin^2^ area, homogeneously observed at 1.2mm. ALCS enables us to detect extremely faint millimeter sources by lensing magnification, including near-infrared (NIR) dark objects showing no counterparts in existing Hubble Space Telescope and Spitzer images. The dust continuum sources belong to a blind sample (N=141) with signal-to-noise ratio (S/N)>~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 1.2mm number counts down to ~7{mu}Jy, and find that the total integrated 1.2mm flux is 20.7_-6.5_^+8.5^Jy/deg^2^, resolving ~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 millimeter emission. We also derive infrared (IR) luminosity functions (LFs) at z=0.6-7.5 with the 1/Vmax 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_-21_^+25^% of the Madau and Dickinson measurements mostly based on rest-frame UV 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 (~60%) SFRD component contributed by the faint millimeter population, including NIR-dark objects.