A simple counterexample for your first question is the uncountable particular point topology (example 10 in Steen and Seebach's Counterexamples in Topology). Let $A$ be an uncountable set, and let $X$ be the disjoint union of $A$ with some additional point $p$. Define a topology on $X$ by taking as open any set which is either empty or contains $p$. Then $X$ is separable (indeed, $\{p\}$ is dense in $X$). Now let $C = \{\{x,p\} : x \in X\}$. This is a basis for the topology of $X$, but for instance, $X$ itself cannot be written as a countable union of sets from $C$. (It's worth noting, though, that $X$ is first countable: the single set $\{x,p\}$ is a neighborhood basis at $x$).
For your second question, one can give a proof using simpler words :) Of course a separable metrizable space is second countable, so has a countable basis $\{U_n\}$. Let $C$ be any basis and $V$ be open. For every $x \in V$, we can find $W_x \in C$ with $x \in W_x \subset V$. We can also find $U_{n(x)}$ with $x \in U_{n(x)} \subset W_x$. We can now choose a countable set $E$ such that $\{n(x) : x \in E\} = \{n(x) : x \in V\}$. (More formally, think of the function $n : V \to \mathbb{N}$; for each $k$, let $E$ contain one element of $n^{-1}(k)$ if it is nonempty.) Now for any $y \in V$, there exists $x \in E$ with $n(x) = n(y)$, so $y \in U_{n(y)} = U_{n(x)} \subset W_x$. Thus $\bigcup_{x \in E} W_x = V$, so $V$ can be written as a countable union of sets from $C$.
Note that this works in any second countable space, metrizable or not.