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!
This is indeed a mystery. I presume the question refers to wedges of spheres of the same dimension, where there's a simple criterion (n-dimensional and (n-1)-connected, for some n). For wedges of spheres of different dimensions I don't know any such criterion. Even when it's known that a complex has the homotopy type of a wedge of n-spheres, it can be difficult to find cycles representing a basis for the homology. Proofs of (n-1)-connectedness are often by induction and hence not really very enlightening, in my experience with complexes arising in combinatorial low-dimensional topology. Ideally such a proof would proceed by showing that after deleting the interiors of some top-dimensional cells, the resulting subcomplex was contractible, hopefully by an explicit contraction. There's even one important case, Harvey's curve complex of a surface, where the complex has higher dimension than the wedge of spheres that it's homotopy equivalent to. It almost seems like a metatheorem in this area that any naturally-defined complex is either contractible or homotopy equivalent to a wedge of spheres. I can't think of any counterexamples, just off the top of my head. Perhaps in other areas the proofs are more enlightening.
Best Answer
There are two papers of M. Wrochna in which ideas from homotopy are used to prove combinatorial theorems:
In http://arxiv.org/abs/1408.2812 he uses ideas from homotopy to give a polynomial-time algorithm for "reconfiguring graph homomorphisms" when the target graph does not contain a $4$-cycle.
In http://arxiv.org/abs/1601.04551 he uses ideas from homotopy to prove a new result on graph homomorphisms which is related to Hedetniemi's Conjecture. Specifically, he proves that all graphs which do not contain a $4$-cycle are "multiplicative." Before this, only one very specific class of graphs was known to be multiplicative (and he also uses homotopy to give a new proof of this result).