Size of BSP tree for a simple polygon?
No, simple polygons are not any easier than disjoint segments. Given any n disjoint segments, there is a simple polygon with O(n) edges that includes all of the input segments as sides of the polygon. Then, any BSP tree for the polygon forms a BSP tree for the segments.

One way to do construct a polygon for a set of segments would be to triangulate the planar straight line graph formed by the input segments, and then pick a spanning tree of the triangulation, which is necessarily non-crossing. Then form a simple polygon by walking around an Euler tour of the spanning tree, using each of the input segments the first time the Euler tour traverses that edge and a parallel segment near it the second time the tour traverses it.