Do we have "higher order graphs" (graphs with vertices defined as edges of other graphs)?

Sure, do what you like. In my graph theory final, we had to analyze the tree graph of a graph $G$, whose vertices are spanning trees of $G$ and whose vertices were connected iff the symmetric difference of their edge sets was a doubleton. So you can make the vertex set of a (finite) graph nearly anything finite that you can describe mathematically.

Sure! This is for example the situation with the Line graph, where each edge of a graph $G$ is a vertex in its line graph $L(G)$, and two vertices of $L(G)$ are connected iff the corresponding edges in $G$ have a common vertex.

This is very useful concept. See this wiki page for details.

One way of thinking about the output of the Szemerédi regularity lemma for a given graph $G$ and a given $\epsilon>0$ is that it gives you a regularity graph for $G$, whose vertices are some collection of subsets $V_1,\cdots,V_k$ of vertices of $G$ and where two such vertices are joined by an edge if they are $\epsilon$-regular. The lemma tells you that all but $\epsilon k^2$ pairs of vertices in this graph are connected.

Then you can apply Turán's theorem to the regularity graph to arrive at a nice proof of the Erdős–Stone theorem.