The compact Hausdorff space $X = \beta\mathbb{N}$ is another example. Every regular open subset is the closure of a subset of $\mathbb{N}$ and there are only $\frak{c}$ such subsets but $X$ has $2^{\frak{c}}$ points, and for each such point $p$, the set $X \setminus \{p\}$ is an open set.
For the second question, a scattered separable space of cardinality $\frak{c}$ and scattered height $2$ (such as a Cantor Tree, or a $\Psi$-space of size $\frak{c}$) gives a space in which every closed subset is a G$_\delta$ but in which the equality fails for the same reason that it fails in the tangent-disk space.