We investigate the onset of nonlinear effects in hybrid polymer-chalcogenide optical microwires and show that they provide an enhanced Kerr nonlinearity while simultaneously mitigating stimulated Brillouin scattering as compared to both chalcogenide and silica optical fibers. It is shown in particular that the polymer cladding surrounding the microwire significantly broadens the Brillouin linewidth and increases the threshold, thus enabling Kerr nonlinear applications. We also study the influence of the wire diameter on the Brillouin dynamics and demonstrate that the Brillouin frequency shift can be finely tuned over a wide radio-frequency range.