Maximum number of geometrically disjoint paths - is the complexity known?
The following paper establishes NP-hardness of essentially all non-trivial questions in this direction:


  Jan Kratochvíl, Anna Lubiw, Jaroslav Nesetril:
  Noncrossing Subgraphs in Topological Layouts. SIAM J. Discrete Math. 4(2): 223-244 (1991)