  We investigate the dynamical interaction between cosmic rays (CRs) and the
multiphase interstellar medium (ISM) using numerical magnetohydrodynamic (MHD)
simulations with a two-moment CR solver and TIGRESS simulations of star-forming
galactic disks. We previously studied transport of CRs within TIGRESS outputs
using a "post-processing" approach, and we now assess the effects of the MHD
backreaction to CR pressure. We confirm our previous conclusion that there are
three quite different regimes of CR transport in multiphase ISM gas, while also
finding that simulations with "live MHD" predict a smoother CR pressure
distribution. The CR pressure near the midplane is comparable to other pressure
components in the gas, but the scale height of CRs is far larger. Next, with a
goal of understanding the role of CRs in driving galactic outflows, we conduct
a set of controlled simulations of the extraplanar region above $z=500$ pc,
with imposed boundary conditions flowing from the midplane into this region. We
explore a range of thermal and kinematic properties for the injected thermal
gas, encompassing both hot, fast-moving outflows, and cooler, slower-moving
outflows. The boundary conditions for CR energy density and flux are scaled
from the supernova rate in the underlying TIGRESS model. Our simulations reveal
that CRs efficiently accelerate extra-planar material if the latter is mostly
warm/warm-hot gas, in which CRs stream at the Alfv\'en speed and the effective
sound speed increases as density decreases. In contrast, CRs have very little
effect on fast, hot outflows where the Alfv\'en speed is small, even when the
injected CR momentum flux exceeds the injected MHD momentum flux.
