Embedding of flat surfaces Let $S$ be a orientable compact surface with a flat euclidean structure with conical singularities (cf. [T] for instance). Let also $\mathcal P$ be a polyhedral euclidean decomposition of $S$ (with vertices at the singular points of the euclidean structure of $S$). 
Question 1:  can $S$ be realized (as a polyhedral surface) in an euclidean space? If yes, what is known about the set of such `euclidean polyhedral realizations'?
Question 2: same questions than above, but for $\mathcal P$ (i.e. an euclidean polyhedral decomposition of $S$ is fixed). 
-- 
Remarks:


*

*there is no assumption  on the genus of $S$ (when $S$ has genus 0, an answer to the (first part of)  Question 1 is  given by Alexandrov theorem)

*A similar question (but for Riemann surfaces) already appears in [B] (page 9). I guess that a lot has been done on this problem since. Any relevant reference would constitute an interesting answer.
Thanks for any help!

[B] Bers L., Riemann surfaces (1958)
[T] Troyanov M. -- Les surfaces a singularités coniques. Enseign. Math. 32 (1986), 79–94.
 A: It may be that this theorem of Burago & Zalgaller (partially) answers your question?

Theorem (Burago-Zalgaller 1.7). 
  Every polyhedron $M$ admits an isometric piecewise-linear $C^0$
  immersion into $\mathbb{R}^3$.
  If $M$ is orientable or has a nonempty
  boundary, then $M$ admits an isometric piecewise-linear $C^0$
  embedding into $\mathbb{R}^3$.

This is from:


*

*Yu. D. Burago and V. A. Zalgaller. 
"Isometric piecewise linear immersions of two-dimensional manifolds with polyhedral metrics into $\mathbb{R}^3$.
St. Petersburg Math. J., 7(3):369--385, 1996. Translated by S. G. Ivanov.
English translation: 
Scanned PDF (15MB).


Here is their definition of a "polyhedron":

By a two-dimensional manifold with polyhedral metric (in brief,
  a polyhedron), we mean a metric space endowed with the structure
  of a connected compact two-dimensional manifold (possibly with
  boundary) every point $x$ of which has a neighborhood isometric to
  the vertex of a cone. ... The metric is locally 
  flat everywhere except
  for a finite collection of points; these points are the `true' vertices."

A nice phrase they use in Lemma 2.2 to describe the mapping is that each
triangle becomes a pleated surface in $\mathbb{R}^3$.
