As far as I understand, I think you have misstated Valiant's result.  

Regarding 1, yes the embedding is assumed to planar, with the edges constrained to follow the 'edges' of the grid.  This is called a *rectilinear embedding*.  Note that only graphs with maximum degree 4 have rectilinear embeddings, hence the degree restriction.  Secondly, the *area* of the embedding is defined to be the area of the smallest box bounding the embedding. Thus, the $O(V)$ area condition is quite strong (in particular, the length or width is $O(\sqrt V)$).  Finally, Valiant's result is actually for *trees* with maximum degree $4$.  He showed that the $O(V)$ area condition is false in general; there are planar graphs with maximum degree 4 that require bounding area $O(V^2)$.