Source to "Strongly zero-dimensional and zero-dimensional is equivalent for separable metrizable spaces"? What is the source of the claim that "Strongly zero-dimensional and zero-dimensional is equivalent for separable metrizable spaces"? My university lecture told me that this is true, but I fail to find it anywhere in literature. Could you provide a source for this?
Definitions
A Hausdorff topological space $X$ is zero dimensional if for every point $x$ of $X$
and every neighborhood $U$ of $x$ in $X$, there exists a nonempty clopen subset $V$ of $X$ such that $x \in V \subset U$.
The clopen basis of any zero-dimensional space is a collection of clopen sets that is closed under complements and finite intersections.
A Hausdorff topological space $X$ is said to be strongly zero-dimensional whenever for every closed subset $A$ of $X$ and every open subset $U$ of $X$ such that $A \subseteq U$, there exists a clopen subset $V$ of $X$ such that $A \subseteq V \subseteq U.$
 A: What you call "strongly zero-dimensional" is usually called ultranormal. In this common notation a Tychonoff space is called strongly zero-dimensional, if its Stone-Cech compactification is zero-dimensional. A space is ultranormal, iff it is normal and strongly zero-dimensional. In particular, for metric spaces, ultranormality is the same as strong zero-dimensionality. See, for instance here.
In the paper cited in the above link, you also find a proof for "Every Lindelof zero-dimensional space is ultraparacompact" (hence ultranormal), thus answering your question, since a separable metric space is, of course, Lindelof. See also Engelking, General topology 6.2.7.
A: It´s easy to see that every strongly zero-dimensional space is a zero-dimensional space, but conversely it is not necessarily true.
Theorem: Every zero-dimensional Lindelöf space is strongly zero-dimensional. You can see this result in Engelking, in the theorems 6.2.2 to 6.2.7.
Answering your question for metrizable spaces. I think you should ask for the space to be metrizable and separable, in order to have a Lindelöf space and use the previous result. I do not have at the moment, a counterexample on a zero-dimensional, metrizable and non-separable space that is not strongly zero-dimensional
