This is a nice problem is which related to a lot of nice mathematics. You are given a 3-dimensional polytope P and you would like to understand the space of S(P) all gepmetric realization of P. The problem is of interest also in higher dimensions. (Two related miracles for 3 dimensional polytopes are the "Koebe-Abdreev-Thurston" circle packing problem and "Steinitz's theorem".) Indeed some proofs of Steinitz's theorem implies that S(P) is a contractible space wose dimension is the number of edges.
while indeed the number of edges is the "right" dimension for this space, this is not obvious: for simplicial polytopes one can relies on "Cauchy's rigidity theorem". Connelly's flexible sphere demonstrates why the degree of freedom argument can fail. Works on rigiditi of polyhedral graphs (Dehn, Alexandrov, and more modern works by Connelly,Whiteley, and many others can be of relevance.)
The question is about an explicit parametrization of S(P). I am not aware of an explicit parametization and description. The works of Sabitov and his school and collaborators are highly relevant. Sabitov's "bellow theorem" ( http://www.emis.ams.org/journals/BAG/vol.38/no.1/1.html ) regarding the invariance of the volume for flexes of simplicial 2-sphere is related to the way the colume of the polytope can be algebraically described by the edge lengths. Sabitov, his students and partners have various additional results even closer to the question but I dont remember right now. (Try also this work by V. Alexander. http://www.springerlink.com/index/J4EXVR63M2QB95PP.pdf /)