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In the Second part of The Konus-Wind Catalog of Gamma-Ray Bursts with Known Redshifts (first part: Tsvetkova et al. 2017; T17), we present the results of a systematic study of gamma-ray bursts (GRBs) with reliable redshift estimates detected simultaneously by the Konus-Wind (KW) experiment (in the waiting mode) and by the Swift/BAT (BAT) telescope during the period from 2005 January to the end of 2018. By taking advantage of the high sensitivity of BAT and the wide spectral band of KW we were able to constrain the peak spectral energies, the broadband energy fluences, and the peak fluxes for the joint KW-BAT sample of 167 weak, relatively soft GRBs (including four short bursts). Based on the GRB redshifts, which span the range $0.04 \leq z \leq 9.4$, we estimate the rest-frame, isotropic-equivalent energy, and peak luminosity. For 14 GRBs with reasonably constrained jet breaks, we provide the collimation-corrected values of the energetics. This work extends the sample of KW GRBs with known redshifts to 338 GRBs, the largest set of cosmological GRBs studied to date over a broad energy band. With the full KW sample, accounting for the instrumental bias, we explore GRB rest-frame properties, including hardness-intensity correlations, GRB luminosity evolution, luminosity and isotropic-energy functions, and the evolution of the GRB formation rate, which we find to be in general agreement with those reported in T17 and other previous studies.