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

Regarding $1$, yes the embedding is assumed to be 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 $O(V^2)$.   

**Edit.**
For the benefit of others who have not followed the chat, here is a summary. The lemma  under consideration is Lemma 2.1 of the paper [Unit Disk Graphs](http://www.sciencedirect.com/science/article/pii/0012365X9090358O) by Clark, Colbourn and Johnson.  The lemma is due to [Valiant](http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf), but a typo was  was introduced by Clark, Colbourn and Johnson.  The $O(|V|)$ should be replaced by $O(|V|^2)$.