@Joel David Hamkins has directly answered the first part of my question by providing a simple counterexample which shows that $\mathbf{Cat}$ does not have the Cantor-Schroder-Bernstein property. It turns out that his counterexample can be extended to completely answer my question. I will now clarify how to approach the second part of my question.
$\mathbf{Pos}$ usually denotes the category of partially ordered sets as objects and monotone functions as morphisms. For any partially ordered set $(A, \leq)$, we may consider its associated categorification $\mathcal{C}(A, \leq)$ (this is simply viewing $(A, \leq)$ as a category).
There is a natural functor $U: \mathbf{Pos} \rightarrow \mathbf{Cat}$ defined such that, for every partially ordered set $X$, $U(X)=\mathcal{C}(X)$ and such that, for every morphism $f$, $U(f)$ is the unique functor for which the object component of $U(f)$ equals $f$.
Now, $U$ is a faithful functor. Thus, to show that there is no bimorphisms from $\mathcal{C}(\mathbb{Q}_{\geq 0}, \leq )$ to $\mathcal{C}(\mathbb{Q}, \leq )$ in $\mathbf{Cat}$, we only need to show that there is no bimorphism from $(\mathbb{Q}_{\geq 0}, \leq )$ to $(\mathbb{Q}, \leq )$ in $\mathbf{Pos}$. But this was done by @Peter LeFanu Lumsdaine in the comments.