Does there exist a countable complete DVR?
A finite integral domain is a field so it would have to be infinite. If we drop completeness $\mathbb{Q}[x]_{(x)}$ would work.
Complete DVRs that come to my mind (rings of integers in finite extensions of $\mathbb{Q}_p$ and the rings of power series in one variable over a field) are uncountable.
The group of topological automorphisms acts transitively on a DVR (translations).
Now, a complete countable metric space has (by Baire) an isolated point.
Thus a complete countable DVR is discrete, so the valuation must be bounded above, so the DVR is a field.