  We investigate the ground state and critical temperature phase diagrams of
the classical and quantum $S=1/2$ pyrochlore lattice with nearest-neighbor
Heisenberg and Dzyaloshinskii-Moriya interactions (DMI). We consider
ferromagnetic and antiferromagnetic Heisenberg exchange as well as direct and
indirect DMI. Classically, three ground states are found: all-in/all-out,
ferromagnetic and a locally ordered $XY$ phase, known as $\Gamma_5$, which
displays an accidental classical U(1) degeneracy. Quantum zero-point energy
fluctuations are found to lift the classical ground state degeneracy and select
the $\psi_3$ state in most parts of the $\Gamma_5$ regime. Likewise, thermal
fluctuations treated classically, select the $\psi_3$ state at $T=0^+$. In
contrast, classical Monte Carlo finds that the system orders at $T_c$ in the
$\psi_2$ state of $\Gamma_5$ for antiferromagnetic Heisenberg exchange and
indirect DMI with a transition from $\psi_2$ to $\psi_3$ at a temperature
$T_{\Gamma_5} <T_c$. The same method finds that the system orders via a single
transition at $T_c$ directly into the $\psi_3$ state for most of the region
with ferromagnetic Heisenberg exchange and indirect DMI. Such ordering behavior
at $T_c$ for the $S=1/2$ quantum model is corroborated by high-temperature
series expansion. To investigate the $T=0$ quantum ground states, we apply the
pseudo-fermion functional renormalization group (PFFRG). The quantum
paramagnetic phase of the pure antiferromagnetic $S=1/2$ Heisenberg model is
found to persist over a finite region in the phase diagram for both direct or
indirect DMI. We find that near the boundary of ferromagnetism and $\Gamma_5$
antiferromagnetism the system may potentially realize a quantum ground state
lacking conventional magnetic order. Otherwise, for the largest portion of the
phase diagram, PFFRG finds the same ordered phases as in the classical model.
