I suspect that there are counterexamples to all three of your proposals. For example, you clarified that the distinguishing neighborhood of a vertex-transitive graph is just its 1-neighborhood. Then the dodecahedral graph and the Desargues graph will have the same decks; in each case you'll just have twenty copies of $K_{1,3}$. Similarly, although I don't have an explicit counterexample offhand, I suspect that if you examine strongly regular graphs with the same degree and number of vertices, you will find plenty of pairs of graphs with the same decks of sub-maximal neighborhoods. (Strongly regular graphs have diameter 2 so again you're just looking at 1-neighborhoods.)
As for your question of what other reconstruction conjectures there are, the most famous is the edge reconstruction conjecture. There is also the vertex-switching reconstruction conjecture and the $k$-reconstruction conjecture (see Bondy's Graph Reconstructor's Manual for definitions). For more examples, you could try contacting Mark Ellingham at Vanderbilt, who keeps a list of papers on the reconstruction conjecture.
There is an interaction between category theory and graph theory in
F.~W.~Lawvere. Qualitative distinctions between some toposes of generalized graphs. In {\em Categories in computer science and logic (Boulder, CO, 1987)/}, volume~92 of {\em Contemp. Math./}, 261--299. Amer. Math. Soc., Providence, RI (1989).
which we have exploited in
R. Brown, I. Morris, J. Shrimpton and C.D. Wensley, `Graphs of Morphisms of Graphs', Electronic Journal of Combinatorics, A1 of Volume 15(1), 2008. 1-28.
But that is actually about possible categories of graphs, which may be the opposite of the question you ask.
If you look at groupoid theory, then "underlying graphs" are fundamental, for example in defining free groupoids. See for example
Higgins, P.~J. Notes on categories and groupoids, Mathematical Studies, Volume~32. Van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1-195.
Groupoids are kind of "group theory + graphs".
Best Answer
There is a very nice survey Local-global phenomena in graphs by N. Linial