  We study $\nu$-Schr\"oder paths, which are Schr\"oder paths which stay weakly
above a given lattice path $\nu$. Some classical bijective and enumerative
results are extended to the $\nu$-setting, including the relationship between
small and large Schr\"oder paths. We introduce two posets of $\nu$-Schr\"oder
objects, namely $\nu$-Schr\"oder paths and trees, and show that they are
isomorphic to the face poset of the $\nu$-associahedron $A_\nu$ introduced by
Ceballos, Padrol and Sarmiento. A consequence of our results is that the
$i$-dimensional faces of $A_\nu$ are indexed by $\nu$-Schr\"oder paths with $i$
diagonal steps, and we obtain a closed-form expression for these Schr\"oder
numbers in the special case when $\nu$ is a `rational' lattice path. Using our
new description of the face poset of $A_\nu$, we apply discrete Morse theory to
show that $A_\nu$ is contractible. This yields one of two proofs presented for
the fact that the Euler characteristic of $A_\nu$ is one. A second proof of
this is obtained via a formula for the $\nu$-Narayana polynomial in terms of
$\nu$-Schr\"oder numbers.
