Reverse-order laws for generalized inverses of matrix products is a classic object of study in the theory of generalized inverses. One of the well-known reverse-order laws for a matrix product AB is (AB)^(i,...,j) = B^(i,...,j)A^(i,...,j), where (·)^i,...,j denotes an {i,...,j}-generalized inverse of matrix. Because {i,...,j}-generalized inverse of a general matrix is not necessarily unique, the relationships between both sides of the reverse-order law can be divided into four situations for consideration. In this article, we first introduce a linear mixed model y = ABβ + Aγ + ε, present two least-squares methodologies to estimate the fixed parameter vector  in the model, and describe the connections between the two least-squares estimators and the reverse-order laws for generalized inverses of the matrix product AB. We then prepare some valued matrix analysis tools, including a general theory on linear or nonlinear matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of B^(i,...,j)A^(i,...,j), as well as necessary and sufficient conditions for B^(i,...,j)A^(i,...,j) to be invariant with respect to the choice of A^(i,...,j) and B^(i,...,j). We then present a unied approach to the 512 set inclusion problems {(AB)^(i,...,j) ⊇ {B^(i,...,j)A^(i,...,j)}for the eight commonly-used types of generalized inverses of A, B, and AB using the block matrix representation method (BMRM), matrix equation method (MEM), and matrix rank method (MRM), where {(·)^(i,...,j)} denotes the collection of all {i,...,j}-generalized inverse of a matrix.