Stable Marriage Problem unique match
I will answer your first question.

By definition, a matching $A$ is male-optimal if for any male $m$ and stable matching $C$, $A(m) \succeq_m C(m)$, that is, $m$ prefers $A(m)$ at least as much as $C(m)$, where $A(m)$ is the female assigned to $m$ by $A$.

Similarly, by definition, a matching $B$ is male-pessimal if for any male $m$ and stable matching $C$, $B(m) \preceq_m C(m)$.

Now suppose that $A$ is male-optimal, $B$ is male-pessimal, and $A = B$. Take any male $m$ and any stable matching $C$. By definition, $A(m) \succeq_m C(m) \succeq_m B(m) = A(m)$, and so $C(m) = A(m)$. In other words, $C = A$, and so the stable matching is unique.