  In this paper, we define 'simplicial GKM orbifold complexes' and `simplicial
GKM graph complexes. We study some of their topological and combinatorial
properties. We discuss some necessary conditions to confirm the invariant
$q$-cell structure on a simplicial GKM orbifold complex. We prove Thom
isomrphism for general orbifold $G$-bundles for equivariant cohomology and
equivariant K-theory with rational coefficients. We use this to extend the main
result of Harada-Henriques-Holm (2005) to the category of $G$-spaces equipped
with `singular invariant stratification'. Then we introduce an equivalent
definition of weighted Grassmann orbifolds as a generalization of Grassmann
manifolds which were discussed by Corti and Reid (2002) and explicitly studied
by Abe and Matsumura (2015) to compute their equivariant cohomology. We study
their different $q$-cell structures and discuss some necessary condition to
determine if they have invariant CW-structures. We employ the techniques from
the first part to study some generalized cohomology theories of weighted
Grassmann orbifolds. We introduce `divisive' weighted Grassmann orbifolds and
compute their equivariant cohomology, equivariant K-theory and equivariant
cobordism ring with integer coefficients. We discuss Schubert calculus for the
equivariant cohomology ring and show how to compute the corresponding structure
constants with integral coefficients.
