Two-dimensional (2D) Janus materials due to their asymmetric structures show fascinating spintronic and piezoelectric properties making them a research hot spot in recent years. In this work, inspired by Janus group-III monochalcogenides, we propose ZnAXY (A = Si, Ge, Sn, and X/Y = S, Se, Te, X < Y ) monolayers as a novel structure with broken inversion and mirror symmetry. By calculating cohesive energy and phonon dispersion, eight of nine possible ZnAXY monolayers are proved to be dynamically stable. In addition, thermal stability of these eight structures is confirmed by ab initio molecular dynamics simulations. The electronic band structures of ZnAXY monolayers indicate that all of them are indirect semiconductors with strong spin-orbit coupling effects. Lack of inversion symmetry gives rise to Zeeman-type spin splitting at the K point of the conduction band with the highest value of 136 meV for ZnSiSeTe. Furthermore, out-of-plane asymmetry results in Rashba spin splitting (RSS) at the F point of the valence and conduction bands in the most compositions of Janus ZnAXY monolayers. Among them, ZnGeSTe with alpha RFV of 1.79 eV A and alpha Fc R of 0.862 eV A and ZnSiSTe with alpha FV R of 1.537 eV A and alpha Fc R of 0.756 eV A are found to be great materials for future spintronic applications. Interestingly, in addition to large RSS, Mexican hat dispersion is observed at the F point of the topmost valence band in these materials. Moreover, the calculated elastic coefficients for the hexagonal ZnAXY monolayers confirm the mechanical stability of the predicted structures. Finally, Janus ZnAXY monolayers possess high in-plane (up to 7.46 pm/V) and out-of-plane (up to 0.67 pm/V) piezoelectric coefficients making them appealing alternatives for prevalent piezoelectric materials.