I have three related questions:


(1) How does one describe the possible Riemannian metrics that are compatible with a conformal structure on a two dimensional surface?


(2) Can all conformable structures be realized through embeddings of the surface in Euclidean 3-space?


(3) How does one understand solutions to the equation


-exp(f) = Laplacian(f), i.e., 
$$- \mathrm{exp} (f) = \Delta f$$


where $f$ is a real-valued function of two variables in an open domain?



metrics compatible with conformal structures
I don't have much to say about (1) or (2), but (3) is, of course, a classical equation and the usual interpretation is this:


I'm assuming that you are talking about a domain in the $z$-plane, also known as the $xy$-plane, and that, by $\Delta f$ you mean the classical $f\_{xx} + f\_{yy}$, not the negative of this or this times some metric scalar (as some sources interpret $\Delta$). In this case, the general solution of $\Delta f = -\exp(f)$ can locally be written in the form


$$
f = \log\left({8\ h'(z)\overline{h'(z)}}\over{(1+h(z)\overline{h(z)})^2}\right)
$$


where $h$ is a (locally defined) holomorphic function on the domain. In the case that the domain is simply connected, one can take $h$ to be globally defined, but you may have to allow it to be meromorphic. 


Geometrically, what is going on is that the conformal metric $ds^2 = \exp(f/2)(dx^2+dy^2)$ has constant curvature $K = 1$ and hence must be constructed by pulling back the standard metric on the $2$-sphere by a conformal map (which one can take to be orientation preserving (in the weak sense), so that it is actually a holomorphic map to the $2$-sphere, thought of as $\mathbb{C}\mathbb{P}^1$ and hence $ds^2$ is the pullback via $h$ of the $K=1$ conformal metric


$$
{{4}\over{(1+w\overline{w})^2}}\ dw\circ d\overline{w}\ .
$$


