Reduction from minimum dominating set to the set cover
Let $G$ be an instance of dominating set and construct an instance of $(U,\mathcal{S})$ of set cover as you describe. The claim is that $G$ has a dominating set of size $k$ if and only if there is a subfamily $\mathcal{S}' \subseteq \mathcal{S}$ with $k$ members that covers $U$.

In the first direction, which you indicate is clearer, let $D$ be a dominating set of size $k$. Construct $\mathcal{S'}$ by adding into it the set corresponding to the closed neighborhood of every $v \in D$. Clearly, $\mathcal{S}'$ has size $k$ as $D$ has size $k$. Further, by the definition of a dominating set, $\mathcal{S}'$ also covers $U$.

In the second direction, let $\mathcal{S}'$ be a set cover for $U$. Build $D$ such that if the set $S_v$, the set corresponding to the closed neighborhood of $v$, is in $\mathcal{S}'$, add the vertex $v$ to $D$. Because $\mathcal{S}'$ covers $U$ which by construction corresponds to the vertex set of $G$, we have that $D$ is a dominating set. Moreover, we clearly have that $D$ has size $k$ as $\mathcal{S}'$ has size $k$.

Your second question however is unclear. A reduction "being related to another reduction" is not a formal statement. In an informal sense, you can use similar ideas to establish other reductions between dominating set, set cover and/or vertex cover but you have to work through the details yourself.