Following up on Simon's comments on 2): in the literature, one of the usual terms for this is that $X$ is *exponentiable*. There is in fact quite a lot of literature on this. Categorically, one is asking that there be a right adjoint $C(X, -)$ to $X \times -: \text{Top} \to \text{Top}$. As it turns out, this is equivalent to the seemingly weaker condition that for the Sierpinski space $Z = \mathbf{2}$ in particular (where the underlying set of $C(X, \mathbf{2})$ is in natural bijection with the topology of $X$), there is a topology so that $C(X, \mathbf{2})$ represents the functor $\hom_{\text{Top}}(X \times -, \mathbf{2})$: there is a natural isomorphism 

$$\hom_{\text{Top}}(-, C(X, \mathbf{2})) \cong \hom_{\text{Top}}(X \times -, \mathbf{2}).$$ 

(Of course the topology on $C(X, \mathbf{2}) \cong \text{Open}(X)$ is uniquely determined by this universal property.) 

Topologically, such $X$ are exactly *core-compact spaces*. To define this notion, write $V\ll U$ to mean that any open cover of $U$ admits a finite subcover of $V$; this is read as "$V$ is relatively compact under $U$" or "$V$ is way below $U$".  We say that $X$ is **core-compact** if for every open neighborhood $U$ of a point $x$, there exists an open neighborhood $V$ of $x$ with $V\ll U$.  In other words, $X$ is core-compact iff for all open subsets $V$, we have $V = \bigcup \{ U | U\ll V \}$. 

This description is synonymous with saying: the topology $\text{Open}(X)$ is a *continuous lattice*, for which there is also a lot of literature, especially in so-called *domain theory* (after Dana Scott). 

Core-compact spaces are a very mild generalization of locally compact spaces (defined here to mean spaces where the neighborhood filter at any point is generated by compact neighborhoods). All locally compact spaces are core-compact. If $X$ is Hausdorff or even merely sober, then $X$ is core-compact iff it is locally compact (along the lines Simon was suggesting, if we consider spatial locales), according to Theorem 8.3.10 in Non-Hausdorff Topology and Domain Theory by Goubault-Larrecq (<a href="https://books.google.com/books?id=QwjAV_t90IcC&pg=PA142&lpg=PA142&dq=core+compact+topology&source=bl&ots=O6ouj3b2De&sig=3azYgBR5e37rnFWwL6kUNgiEHwQ&hl=en&sa=X&ei=Ak_pUt3zEqGMyQGmqIDADw#v=onepage&q=core%20compact%20topology&f=false">partially viewable</a> in Google Books).  

It seems to be surprisingly hard to cook up (or even find in the literature!) a core-compact space that is not locally compact, but you can find such an example in <a href="http://www.ams.org/journals/tran/1978-246-00/S0002-9947-1978-0515540-7/S0002-9947-1978-0515540-7.pdf">this paper</a> by Hofmann and Lawson (section 7). 

Another fine compendium of results is <a href="http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf">Topologies on Spaces of Continuous Functions</a> by Escard&oacute; and Heckmann (I believe it addresses aspects of your other <a href="https://mathoverflow.net/questions/242457/admissible-and-proper-topologies-on-cx-y">recent question</a> on admissible topologies on function spaces, but I've not had time to look into this carefully.) 

As for 1), I'm currently somewhat skeptical that a nice answer can be given, but here again I've not had time to investigate this properly.