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A possibly interesting variant of the question: Given an arbitrary infinite set of primes P, is there some N, so that for every n > N, A_n is generated by two elements with orders in P?
Doesn't the sphericity of (3) follow from the following easier observation? A set of rank(L) elements in a geometric lattice generates a Boolean lattice, so (passing to the order complex) a subdivision of the boundary of a simplex.
The homology contributed package used to have some routines to deal with (full) coset lattices of finite groups, but is difficult to get going on a modern machine. The simpcomp package will compute homology, given say a list of facets. Finding the list of facets of an order complex of a poset defined from a group looks like something that you should be able to reasonably do with GAP!
I'll also comment that there's little reason to require pure-ness here. For vertex-decomposable, this goes back to Björner and Wachs. The "pure deletion" condition in the general situation becomes a sensible exchange condition. My favorite place to read about this is my paper :-) "Chordal and sequentially Cohen-Macaulay clutters"; you can also find it in Jonsson's "Optimal decision trees on simplicial complexes".
@user177523, my point is that any shellable but not vertex-decomposable complex is a counterexample to the thing you expect to prove. Take any last step where there is no shedding vertex. Explicit example of shellable but not vertex-decomposable can be found in arxiv.org/abs/1505.02837 . (The example there is even flag.)
Mayer-Vietoris tells you how to compute the homology of the union of two simplicial complexes. That's pretty combinatorial! The details do involve some linear algebra, as you'd expect for a linear algebraic invariant.
As far as combinatorial versions of homotopy equivalence: you might like collapsibility and its ilk. See en.wikipedia.org/wiki/Collapse_(topology) . This leads eventually towards discrete Morse theory, in which one alternatively collapses and "sets faces aside" (while remembering what they attach to).
There are plenty of sufficient (or even necessary and sufficient) criteria for C to be a matroid. Pretty much any of these will directly translate over the Alexander duality barrier to a criterion for D(C) to be a matroid. I'll mention that a specific place where the situation you mention has arisen is in the paper of Eagon and Reiner "Resolutions of Stanley-Reisner rings and Alexander duality".
LyX generates fairly reasonable LaTeX, which I like, and is a relatively thin layer over LaTeX, which I also like. I've had very little trouble submitting papers to journals. Have you tried it with a LibreOffice LaTeX export? Collaborators are indeed the rub.
