The far side of the Milky Way's disk is one of the most concealed parts of the known universe due to extremely high interstellar extinction and point-source density toward low Galactic latitudes. Large time-domain photometric surveys operating in the near-infrared hold great potential for the exploration of these vast uncharted areas of our Galaxy. We conducted a census of distant classical and type II Cepheids along the southern Galactic midplane using near-infrared photometry from the VISTA Variables in the Via Lactea survey. We performed a machine-learned classification of the Cepheids based on their infrared light curves using a convolutional neural network. We have discovered 640 distant classical Cepheids with up to ~40mag of visual extinction and over 500 type II Cepheids, most of them located in the inner bulge. Intrinsic color indices of individual Cepheids were predicted from sparse photometric data using a neural network, allowing their use as accurate reddening tracers. They revealed a steep, spatially varying near-infrared extinction curve toward the inner bulge. Type II Cepheids in the Galactic bulge were also employed to measure robust mean selective-to-absolute extinction ratios. They trace a centrally concentrated spatial distribution of the old bulge population with a slight elongation, consistent with earlier results from RR Lyrae stars. Likewise, the classical Cepheids were utilized to trace the Galactic warp and various substructures of the Galactic disk and uncover significant vertical and radial age gradients of the thin disk population at the far side of the Milky Way.