For your question #5, this can be generalized to: given two categories $A$ and $B$, describe the functors $A\to B$. For a closer match, you can restrict to the functors which map irreducible maps (those that cannot be written as a composition of two non-identity maps) to irreducible maps.
You example corresponds to letting $A$ be the poset you drew considered as a category, and $B$ the category of graphs and embeddings.
You can generalize a bit differently by asking: given a category $A$, describe the pairs $(B,F)$ with $B$ a category and $F:A\to B$ a functor (preserving irreducibility of maps, if you want...). It would not be without interest to know of another such functor from your poset to a category which is not graphs.