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If I understand what you're looking for correctly, it suffices to delete the faces consisting of one point from every $X_i$. Since one can specify a simplicial complex either by the maximal faces or else by the minimal non-faces, this complex is well-defined. (You're just adding some new minimal non-faces to the join.) The induced subcomplex on such a vertex subset (consisting of one point from each complex) will be a simplex boundary, hence a sphere. I haven't seen these studied anywhere, but that doesn't mean that they haven't been.
It's been long enough that you've probably either solved the problem or lost interest, but since no one else has mentioned it, I'll point out that the property you mention is almost that of semimodularity. That is, if you require the diamond and disallow the pentagon (in your two allowed diagrams given two elements covering a common element) it is exactly semimodularity. Semimodularity is well-studied -- see e.g. the book by Manfred Stern.
I'll mention for general interest that Eran Nevo studied edge contractions in his thesis, which is available on the arXiv: arxiv.org/pdf/0709.3265.pdf . He says more about the consequences of an edge contraction than the existence of one, but you might find the algebraic shifting arguments in there interesting.
@Igor: The cover letter is where you get to tell the story of why you'd fit well into this particular job. (Or at least to show that you've read the advertisement.) That's what Alexander is talking about, I think. I didn't like jumping through that hoop when I was on the market, but ignoring it is not advised. I have heard of other schools using the cover letter as the basis for an early winnowing.
For the SLLN, isn't there an issue in exchanging the limit with the integral implicit in the variance? Even for the WLLN, it seems to me that the technical details needed (e.g. Chebyshev's Inequality) rather "ruin the pristine elegance", to quote a previous comment.
Another reference which looks fairly related is Ziegler's paper "Shelling polyhedral 3-balls and 4-polytopes". (Likely you've already seen this, although you don't mention it in your question.) He's mainly looking at extendable shellability in there, however.
Ok, so my prior attempt missed a condition, but the answer still seems to me like it should be "no". Won't the group algebra of (Z_p)^2 over the field Z_p do as a counterexample? According to Wikipedia this is a local ring. It has at least 3 subalgebras of dimension 2 (corresponding to subgroups of order p), whose meet is Z_p, and whose join is the whole algebra. (And this contradicts distributivity.)
