Is there a suitable algorithm to draw a mixed constituency/dependency graph in a coordinate system?
you seem to want the following


a graph layout algorithm. many std algorithms typically use force-based methods where nodes are connected by replusive/attractive "springs" and the equilibrium/low energy state is obtained through some amt of iteration (of the corresponding globally-constructed force-related differential equation). the output of the algorithm is a set of 2-d or 3-d coordinates for nodes and (usually) edges.
ability to hand-modify the results of that algorithm. typically called a "graph editor". this starts with the coordinates from the layout algorithm and allows adjustment.


re (1) the forefront software seems to be graphviz. re (2) see eg this question "what is the best graph editor" on mathoverflow.

browsing the graphviz gallery, here are two graph types similar to what you want. 

ER diagram and traffic light. 

you say you have two types of edges. a simple way would be to have directed edges "toward or away" as in the traffic light example. or the edges can be labelled in two ways as in the ER diagram. both examples show two different node types using different shapes or labels, or shading etc. other approaches would be to use coloring.

as the mathoverflow Q/As indicate there are many graph editors. the std one with graphviz is "dotty". see eg pdf "editing graphs with dotty" by Koutsofious.

another technique for graphing, possibly large graphs, is to look at structural compositions eg clique decompositions.