Explain phrase "For each pair of distinct elements" in reference to a graph.

Also, "connected" does not mean by a directed edge, but by a directed path.

I think "for each pair of distinct vertices $(v_i, v_j)$ in $S \times S$" means

"for each $(v_i, v_j) \in S \times S$, $v_i \neq v_j$."

From your subset $S$ of vertices you can construct a set of pairs $P$, where $(v,w) \in P$ if there is a directed path from $v$ to $w$ (not necessarily confined to $S$). The set is strongly connected if $$ \forall v,w\in S, ( v\neq w \Rightarrow (v,w) \in P \wedge (w,v) \in P) $$ Let $S=G$ in the example graph. Notice that $(A,B)$, $(B,C)$, $(C,D)$, $(D,E)$, $(E,A)$ are all in $P$. So by transitivity, every pair of distinct vertices is in $P$.

The word distinct in the definition rules out the requirement that $(v,v) \in P$. However, it seems that $(v,w) \in P \wedge (w,v) \in P$ would imply that $(v,v)\in P$ anyway.

I'm not sure what you mean by “valid” pair. Any pair may be in $P$; the condition is only that certain pairs must be in $P$.