[Added: This is a follow-up of an earlier post.]

Consider the following "reconstruction puzzle", stated informally:

Given a concrete poset, e.g. the poset of undirected unlabeled finite graphs without isolated vertices, ordered by embeddability (arrow heads, identities and composition omitted in the diagram):

Now forget about the inner structure of the objects and consider only the corresponding abstract poset:

The "reconstruction puzzle" is to reconstruct the inner structure of the objects unambiguously from their "positions" in the poset.

It's obvious what a solution of this puzzle is and that it can be solved "by hand" for at least some of the smaller objects, just carefully considering the in- **and** out-arrows.

Question #1: How can such a puzzle be stated formally?

Question #2: How can it be solved "algorithmically"?

Question #3: What's the mathematics behind this kind of puzzle?

I suppose it's not category theory, since category theory is not concerned with the inner structure of objects.

Question #4: Can it be shown - at least for this concrete example - that every object is reconstructible up to isomorphism?

Since for this concrete example there are no isomorphic objects we could have omitted "up to isomorphism".

Question #5: What's an obvious way to generalize this kind of puzzles (from posets to what?)