Planar Embedding with Some Nodes Constrained
This was meant to be a comment but it was a bit too long, sorry!

There is a well known algorithm to draw a planar graph, namely Tutte's drawing algorithm. The input graph is assumed to be 3-connected and planar. The idea of the algorithm is to fix the position of vertices of a face in convex position and from those coordinates deduce the positions for the rest of the vertices. The resulting drawing being a planar drawing of the input graph. Perhaps you can find more about this at [1].

With respect to grids, there are a couple of well known algorithms for drawing planar 3-connected graphs. One is Schnyder's algorithm (through the a Schnyder wood decomposition of the edges), and the canonical ordering algorithm (through a partition of the vertices of the graph). Both of these algorithms run in $O(n)$ time and produce drawings in an $O(n)\times O(n)$grid, where $n$ is the number of vertices of the input graph. At [2] you may find an overview of these two algorithms for the case of planar triangulations. You could even go to references [4] and [9] cited there for the original papers. It is worth noting that these algorithms have been generalized to work on 3-connected planar graphs as stated above.

Are you thinking about drawings subject to more specific constraints?