Is $c_0$ a $C(K)$ space? Is $c_0$ isomorphic or isometric so $C(K)$ for a countable metric space $K$?
 A: The question doesn't specify what sort of isomorphism we want.
Also it's not specified exactly what we mean by $C(K)$ if $K$ is not compact. It seems likely that however we interpret $C(K)$ here it contains the constants, hence it cannot be isomorphic to $c_0$ as a complex algebra, since $c_0$ has no identity. Various comments about $C^*$ algebras are wrong, since $c_0$ has no identity; the most we can get from that context is $C_0(K)$, where $K$ is locally compact, and of course that's of no interest since $c_0$ is equal to $C_0(\Bbb N)$.
It appears to me that $c_0$ is Banach-space-isomorphic to $C(K)$ with $K$ compact. Let $\approx$ denote Banach-space isomorphism.
Let $K=\Bbb N\cup\{\infty\}$ be the one-point compactification of $\Bbb N$. Then $$c_0\approx\{f\in C(K):f(\infty)=0\}:=X,$$so $C(K)\approx \Bbb C\times c_0.$ (Because $(\alpha,f)\mapsto f+\alpha$ gives an isomorphism of $\Bbb C\times X$ onto $C(K)$.)
And $(\alpha,(x_1,x_2,\dots))\mapsto(\alpha,x_1,x_2,\dots)$ shows that $$\Bbb C\times c_0\approx c_0.$$
