Several questions around the exponential law

As to Question 2, let $\Sigma$ be the composition map $\mathrm{Map}(X,Y)\times\mathrm{Map}(Y,Z)\rightarrow\mathrm{Map}(X,Z)$.

All functions spaces have the compact-open topology, with as a subbase all sets of the form $\mathrm{M}(C,U) = \{ f: f[C] \subset U \}$, where $C$ is a compact subset of the domain, and $U$ an open subset of the co-domain.

Let $(f,g)$ be a point in $\Sigma^{-1}[\mathrm{M}(C,U)]$, with $C \subset X$ compact and $U \subset Z $ open, and we want to show it's an interior point. We have by definition $(g \circ f)[C] \subset U$, or $f[C] \subset g^{-1}[U]$. As $g^{-1}[U]$ is open, and $f[C]$ is compact, both in $Y$, and if we assume $Y$ is locally compact (in the sense from the comments), we can find for each $y \in f[C]$ a compact neighbourhood $K_y$ that sits inside $g^{-1}[U]$, and so $f[C]$ is covered by finitely many sets of the form $\mathrm{Int}(K_y)$, say $\mathrm{Int}(K_{y_i})$ for $i=1 \ldots n$ and set $W$ to be the finite union of these. Then $W \subset \mathrm{Int}(\cup_{i=1}^n K_{y_i}) \subset K:=\cup_{i=1}^n K_{y_i} \subset g^{-1}[U]$ and so $\Sigma[\mathrm{M}(C,W) \times \mathrm{M}(K,U) ] \subset \mathrm{M}(C,U)$ and so we are done, as $(f,g)$ is in $\mathrm{M}(C,W) \times \mathrm{M}(K,U)$.

This convinces me that indeed we only need local compactness (in a rather strong sense) on $Y$ to show continuity of $\Sigma$, but no Hausdorffness.