We have performed 13 three-dimensional global magnetohydrodynamic (MHD) simulations of the magnetosheath plasma and magnetic field properties for Parker spiral (PS) and ortho-Parker spiral interplanetary magnetic field (IMF) orientations corresponding to a wide range of solar wind plasma conditions. To study the growth of the Kelvin-Helmholtz instability on the dawn and dusk flank magnetopause, we have performed 26 local two-dimensional MHD simulations, with the initial conditions taken from global simulations on both sides of the velocity shear layer at the dawn-dusk terminator. These simulations indicate that while the MHD physics of the fast shocks does not directly lead to strong asymmetry of the magnetosheath temperature for typical solar wind conditions, the magnetosheath on the quasi-parallel shock side has a smaller tangential magnetic field along the magnetosheath flow which enables faster growth of the Kelvin-Helmholtz instability (KHI). Because the IMF is statistically mostly in the PS orientation, the KHI formation may statistically favor the dawnside flank. For all the 26 simulations, the growth rates of the KHI correlated well with the ratio of the velocity shear and Alfvén speed along the wave vector, k. Dynamics of the KHI may subsequently lead to formation of kinetic Alfvén waves and reconnection in the Kelvin-Helmholtz vortices which can lead to particle energization. This may partly help to explain the observed plasma sheet asymmetry of cold-component ions, which are heated more on the dawnside plasma sheet.