  We consider the disordered monomer-dimer model on cylinder graphs
$\mathcal{G}_n$, i.e., graphs given by the Cartesian product of the line graph
on $n$ vertices, and a deterministic graph. The edges carry i.i.d. random
weights, and the vertices also carry i.i.d. random weights, not necessarily
from the same distribution. Given the random weights, we define a Gibbs measure
on the space of monomer-dimer configurations on $\mathcal{G}_n$. We show that
the associated free energy converges to a limit, and with suitable scaling and
centering, satisfies a central limit theorem. We also show that the number of
monomers in a typical configuration satisfies a law of large numbers and a
central limit theorem with appropriate centering and scaling. Finally, for an
appropriate height function associated with a matching, we show convergence to
a limiting function and prove the Brownian motion limit about the limiting
height function in the sense of finite-dimensional distributions.
