We use analytic calculations and time-dependent spherically-symmetric simulations to study the properties of isothermal galactic winds driven by cosmic-rays (CRs) streaming at the Alfven velocity. The simulations produce time-dependent flows permeated by strong shocks; we identify a new linear instability of sound waves that sources these shocks. The shocks substantially modify the wind dynamics, invalidating previous steady state models: the CR pressure $p_c$ has a staircase-like structure with $dp_c/dr simeq 0$ in most of the volume, and the time-averaged CR energetics are in many cases better approximated by $p_c propto rho^{1/2}$, rather than the canonical $p_c propto rho^{2/3}$. Accounting for this change in CR energetics, we analytically derive new expressions for the mass-loss rate, momentum flux, wind speed, and wind kinetic power in galactic winds driven by CR streaming. We show that streaming CRs are ineffective at directly driving cold gas out of galaxies, though CR-driven winds in hotter ISM phases may entrain cool gas. For the same physical conditions, diffusive CR transport (Paper I) yields mass-loss rates that are a few-100 times larger than streaming transport, and asymptotic wind powers that are a factor of $simeq 4$ larger. We discuss the implications of our results for galactic wind theory and observations; strong shocks driven by CR-streaming-induced instabilities produce gas with a wide range of densities and temperatures, consistent with the multiphase nature of observed winds. We also quantify the applicability of the isothermal gas approximation for modeling streaming CRs and highlight the need for calculations with more realistic thermodynamics.
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