the best approach to the geometric euler characteristic comes from the theory of o-minimal structures.
the best reference in this area is the book "tame topology and o-minimal structures" by lou van den dries. requires very little background to understand.
in brief: an o-minimal structure is collection of boolean algebras of subsets of $R^n$ which satisfies a short list of axioms. (the name comes from model theory, but you don't need to know any model theory to understand the results)
examples of o-minimal structures include the semialgebraic sets, the globally subanalytic sets, and (if you tweak the definitions a bit) the piecewise-linear sets.
elements of an o-minimal structure are "tame" or "definable" sets. mappings between tame sets are tame iff their graph is a tame set.
basic relevant results:
every tame set has a well-defined euler characteristic.
two tame sets are "definably homeomorphic" (there is a tame bijection between them --- not necessarily continuous!) iff they have the same dimension and euler characteristic.
(yes, i wrote iff - this is the first surprise in this subject)
one can so this for more general manifolds as well.
concerning integration with respect to euler characteristic:
1) in the o-minimal framework, one can integrate all constructible functions, as noted by viro and schapira in the 1980s, based on works of macpherson and kashiwara in the 1970s. these results follow from sheaf theory. though more difficult than the combinatorial approach, all these proofs are "natural" and don't rely on "luck".
2) if you want to integrate non-constructible (e.g., smooth) integrands, the theory of chen (really due to rota) will fail -- that integral vanishes on all continuous integrands.
3) baryshnikov and ghrist have extensions of the integral to definable integrands (see 2009 arxiv paper). there are two such extensions, and they are dual. there are deep connections with morse theory, but the integral operators are unfortunately non-linear, and the fubini theorem does not hold in full generality.
This is a very interesting (yet rather vague) question. Most answers were in the direction of mathematical logic but I am not sure this is the only (or even the most appropriate) way to think about it. The notion of coincidence is by itself very complicated. (See http://en.wikipedia.org/wiki/Coincidence ). One way to put it on rigurous grounds is using probabilistic/statistical framework. Indeed, as Timothy mentioned it is sometimes possible to give a probabilistic heuristic in support of some mathematical statement. But its is notorious statistical problem to try to determine aposteriori if some events represent a coincidence.
I am not sure that (as the OP assumes) if a statement is "true by accident" it implies that it can never be proved. Also I am not sure (as implied by most answers) that "can never be proved" should be interpreted as "does not follow from the axioms". It can also refers to situations where the statement admits a proof, but the proof is also "accidental" as the original statement is, so it is unlikely to be found in the systematic way mathematics is developed.
In a sense (as mentioned in quid's answer), the notion of "true by accident" is related to mathematics psychology. It is more related to the way we precieve mathematical truths than to some objective facts about them.
Regarding the reconstruction conjecture. Note that we can ask if the conjecture is true for graphs with at most million vertices. Here, if true it is certainly provable. So the logic issues disappear but the main issue of the question remains. (We can replace the logic distinctions by computational complexity distinctions. But still I am not sure this will cpature the essence of the question.) There is a weaker form of the conjecture called the edge reconstruction conjecture (same problem but you delete edges rather than vertices) where much is known. There is a very conceptual proof that every graph with n vertices and more than nlogn edges is edge-reconstructible. So this gives some support to the feeling that maybe vertex reconstruction can also be dealt with.
Finally I am not aware of a heuristic argument that "there would need to be a massive coincidence for two non-isomorphic graphs to have the same 'deck'" as the OP suggested. (Coming up with a convincing such heuristic would be intereting.) It is known that various graph invariants must have the same value on such two graphs.
Best Answer
I passed on your question to John H. Conway. Here is his response: (NB. Everything following this line is from Conway and is written from his point of view. Of course, in the comments and elsewhere on the site, I am not Conway.)
I think it's wrong to focus on block designs in particular. This may not answer your question, but there are some interesting examples of theorems similar to Desargues's and Pappus's theorems. They aren't block designs, but they do have very nice symmetries.
I call these "presque partout propositions" (p.p.p. for short) from the French "almost all". This used to be used commonly instead of "almost everywhere" (so one would write "p.p." instead of "a.e."). The common theme of the propositions is that there is some underlying graph, where vertices represent some objects (say, lines or points) and the edges represent some relation (say, incidence). Then the theorems say that if you have all but one edge of a certain graph, then you have the last edge, too. Here are five such examples:
Desargues' theorem
Graph: the Desargues graph = the bipartite double cover of the Petersen graph
Vertices: represent either points or lines
Edges: incidence
Statement: If you have ten points and ten lines that are incident in all of the ways that the Desargues graph indicates except one incidence, then you have the last incidence as well. This can be seen to be equivalent to the usual statement of Desargues's theorem.
Pappus's theorem
Graph: the Pappus graph, a highly symmetric, bipartite, cubic graph on 18 vertices
Vertices: points or lines
Edges: incidence
Statement: Same as in Desargues's theorem.
"Right-angled hexagons theorem"
Graph: the Petersen graph itself
Vertices: lines in 3-space
Edges: the two lines intersect at right angles
Statement: Same as before, namely having all but one edge implies the existence of the latter. An equivalent version is the following: suppose you have a "right-angled hexagon" in 3-space, that is, six lines that cyclically meet at right angles. Suppose that they are otherwise in fairly generic position, e.g., opposite edges of the hexagon are skew lines. Then take these three pairs of opposite edges and draw their common perpendiculars (this is unique for skew lines). These three lines have a common perpendicular themselves.
Roger Penrose's "conic cube" theorem
Graph: the cube graph Q3
Vertices: conics in the plane
Edges: two conics that are doubly tangent
Statement: Same as before. Note that this theorem is not published anywhere.
Standard algebraic examples
Graph: this unfortunately isn't quite best seen as a graph
Statement: Conics that go through 8 common points go through a 9th common point. Quadric surfaces through 7 points go through an 8th (or whatever the right number is).
Anyway, I don't know of any more examples.
Also, I don't know what more theorems one could really have about coordinatization. I mean, after you have a field, what more could you want other than, say, its characteristic? (Incidentally, the best reference I know for the coordinatization theorems is H. F. Baker's book "Principles of Geometry".)
In any case, enjoy!