  We study the random conductance model on the lattice $\mathbb{Z}^d$, i.e. we
consider a linear, finite-difference, divergence-form operator with random
coefficients and the associated random walk under random conductances. We allow
the conductances to be unbounded and degenerate elliptic, but they need to
satisfy a strong moment condition and a quantified ergodicity assumption in
form of a spectral gap estimate. As a main result we obtain in dimension $d\geq
3$ quantitative central limit theorems for the random walk in form of a
Berry-Esseen estimate with speed $t^{-\frac 1 5+\varepsilon}$ for $d\geq 4$ and
$t^{-\frac{1}{10}+\varepsilon}$ for $d=3$. Additionally, in the uniformly
elliptic case in low dimensions $d=2,3$ we improve the rate in a quantitative
Berry-Esseen theorem recently obtained by Mourrat. As a central analytic
ingredient, for $d\geq 3$ we establish near-optimal decay estimates on the
semigroup associated with the environment process. These estimates also play a
central role in quantitative stochastic homogenization and extend some recent
results by Gloria, Otto and the second author to the degenerate elliptic case.
