Well, filter first on the order condition and product condition (which are quicker to check) then on the generation condition. Unless I'm mistaken, you might as well require the first entry to have the highest order. What have you tried?

I'd generally interpret being a subdivision as meaning "arising by repeated stellar subdivision." For example, the barycentric subdivision arises by iteratively subdividing all faces in a linear extension of reverse inclusion. OTOH, Stanley in Combinatorics and commutative algebra has an alternative that you might like more: he defines a subdivision of simplicial complex \Delta to be a complex \Delta' s.t. each face of \Delta' is contained in a face of \Delta, where his inclusion comes from a geometric realization. Should be easy to abstract out usefully, I think.

Are you looking to detect whether K is obtained from L by repeated subdivisions computationally? Or in what sense do you want to detect subdivisions? Another term that might be along the lines of what you are looking for is "bistellar flip", although this is not in general straightforwardly a subdivision. But I'll mention that Frank Lutz and others have done computational work on bistellar flips for manifolds.

Note that the local homology can be recovered (after a shift) from the homology of the links. See Munkres' "Topological results in combinatorics". I haven't checked carefully, but I think the condition in the question reduces to finding a complex where the link of each vertex has nontrivial homology, and the link of each higher face has \mathbb{Z} in the top homology degree, 0 elsewhere.

To clarify one detail in the comment of Theo Johnson-Freyd and connect with the comment of Geoff Robinson: one also wants the permutation representation to be faithful, which corresponds to the subgroup $H$ being core-free. Of course, this is trivial in a simple group.

In the direction of a zoo: it's sporadically updated, but have you seen eg-models.de/models_noapplet.html ? It has many small triangulations of various nice spaces, with pictures + data.

@LSpice, I think I put Tenner first because I was replying to Timothy Chow's comment on Tenner's talk. (But you're right, I should've used the canonical order.)

To clarify this comment (much later), @AntonPetrunin is using a description of join for locally compact Hausdorff spaces given in an exercise for Lecture 2 of the book "Homotopical Topology" (and probably elsewhere). It describe the join as the subspace of the product of cones where the "height" in the cones sums to 1.