To QUESTION 1: Yes, exponential law can be established without Hausdorff property of the middle space.
In fact the problem consists of four different parts each requiring its own arguments. Let me try to clarify the points.
For sets $ M, N $ denote by $Map( M, N )$ the set of all maps $ M\to N $.
Let $ L $ be another set.
Then there is a bijection
$$ \exp:Map( L \times M, N ) \to Map( L , Map( M, N ))$$
given by the following rule: if $ F \in Map( L \times M, N )$, then $\exp( F ): L \to Map( M, N )$ is given by $\exp( F )(t)(x) = F (t,x)$.
Lemma.
Let $ L , M, N $ be topological spaces.
Then the following statements hold.
(1)
$\exp(C(L \times M, N)) \subset C(L,C(M,N))$, i.e. if $F: L \times M \to N$ is continuous, then $\exp(F): L \to C(M,N)$ is continuous as well.
(2)
The induced map
$$ \exp:C(L \times M, N) \to C(L,C(M,N)) $$
is continuous with respect to the corresponding compact open topologies.
(3)
Suppose $ M$ is locally compact in the sense that for each $x\in M$ and an open neighborhood $ U $ of $x$ there exists a compact subset $ B \subset M$ such that $x\in Int{ B } \subset B \subset U $, (notice that it is not required that $ M$ is also Hausdorff).
Then
$$ \exp (C(L \times M, N)) = C(L,C(M,N)),$$
i.e. the above map is a continuous bijection.
(4)
If in addition to (3) $ L $ is a $T_3$-space (again not necessarily Hausdorff) then the map
$$ \exp: C(L \times M, N) \to C(L,C(M,N)) $$
is a homeomorphism.
Proof.
(1)
Let $ F : L \times M \to N $ be a continuous map.
We need to prove that $ f = \exp( F ): L \to C(M,N)$ given by $ f (t)(x) = F (t,x)$ is continuous.
Let $t\in L $ and
$$
[ B , W ] = \{ g \in C(M,N) \mid g( B )\subset W \}
$$
be an open neighborhood of $ f (t)$ in $C(M,N)$.
We need to find an open neighborhood $ U $ of $t$ in $ L $ such that $ f ( U ) \subset [ B , W ]$
This means that $ f (t)( B ) \subset W $, that is $t\times B \subset F ^{-1}( W )$.
Since $ B $ is compact, there exists an open neighborhood $ U $ of $t$ in $ L $ such that $ U \times B \subset F ^{-1}( W )$.
But this means that for each $s\in U $, $ f (s)( B )\subset W $, i.e. that $ f ( U ) \subset [ B , W ]$.
(2)
Let $ F : L \times M \to N $ be a continuous map and $ f = \exp( F ): L \to C(M,N)$ is given by $ f (t)(x) = F (t,x)$ is continuous.
Let also $A \subset L $ and $ B \subset M$ be compact subsets, $ W \subset N $ be an open set, and
$$
[A, [ B , W ]] = \{
g: L \to C(M,N) \mid
g(A)( B ) \subset W
\}
$$
be an open neighborhood of $ f $ in $C(L,C(M,N))$.
Then $[A\times B , W ]$ is an open neighborhood of $ F $ in $C(L \times M, N)$ such that
$$ \exp([A\times B , W ]) \subset [A, [ B , W ]]. $$
(3)
Suppose $ M$ is locally compact in the above sense and let $ f : L \to C(M,N)$ be a continuous map.
We should prove that $ F = \exp^{-1}( f ): L \times M \to N $, $ F (t,x)= f (t)(x)$ is continuous.
Let $(t,x)\in L \times M$ and $ W $ be an open neighborhood of $ F (t,x)$ in $ N $.
We need to find neighborhoods $ U \subset L $ and $V \subset M$ of $t$ and $x$ such that $ F ( U \times V)\subset W $.
Since $ f (t): M\to N $ is continuous, $ f (t)^{-1}( W )$ is an open neighborhood of $x$ in $ M$, whence there exists a compact $ B \subset M$ such that $x\in Int{ B } \subset B \subset U $.
Then $[ B , W ]$ is an open neighborhood of $ f (t)$, and since $ f : L \to C(M,N)$ is continuous at $t$, there exists an open neighborhood $ U \subset L $ of $t$ such that $ f ( U )\subset [ B , W ]$.
This implies that $ F ( U \times B ) \subset W $.
In particular, one can put $V= Int{ B }$, then we will have $ F ( U \times V) \subset W $.
(4)
Suppose $ L $ is a $T_3$ space, i.e. for each point $x\in L $ and its open neighborhood $ U $ there exists an open neighborhood $ Z$ such that $x\in Z\subset\overline{ Z} \subset U $, and $ M$ is locally compact as in (3).
Let $ F \in C(L \times M, N)$, $ f = \exp( F )$, and $[C, W ]$ be an open neighborhood of $ F $, where $C\subset L \times M$ be a compact subset and $ W \subset N $ is open.
We need to prove that $\exp([C, W ])$ contains an open neighborhood of $ f $ in $C(L,C(M,N))$.
Since $C$ is compact and $ F ^{-1}( W )$ is its open neighborhood in $ L \times M$, there exist finitely many open sets $ U _1,\ldots, U _k \subset L $, and $V_1,\ldots,V_k \subset M$, such that
$$
C \subset \cup_{i=1}^{k} U _i \times V_i \subset F ^{-1}( W ).
$$
Moreover, since $ L $ is $T_3$ and $ M$ is locally compact, one can assume in addition that
(a) each $V_i = Int{ B _i}$ for some compact $ B _i \subset M$;
(b) and that
$$
C \ \subset\ \cup_{i=1}^{k} U _i \times V_i
\ \subset \
\cup_{i=1}^{k} \overline{ U _i} \times B _i
\ \subset \
F ^{-1}( W ).
$$
Let $p: L \times M\to L $, $p(t,x)=t$, be the natural projection.
Then $p(C)$ is compact.
Hence $A_i := \overline{ U _i} \cap p(C)$ is a closed subset of $p(C)$ and therefore compact.
It then follows that
$$ C \subset
\cup_{i=1}^{k} A_i \times B _i \subset F ^{-1}( W ).
$$
This also implies that
$$
\cup_{i=1}^{k} [A_i, [ B _i, W ]] \ \subset \exp( [C, W ] ).
$$
This proves that $\exp$ is open and therefore a homeomorphism.
Thus for the proof of (1) and (2) we do not need any assumptions on $L$ and $M$; for (3) we need local compactness of $M$ and for (4) we need an additional $T_3$ property of $L$.
