You could try a two part approach; on the straight edges you might try the planar brush with a large brush size. For the curve you can use the clip circle brush.
You can try ReMesh All and then Project All; ZBrush seems to ignore the artifacts from clipping.
To cut up the sphere like the Photoshop image:
Hide portions of the mesh then poly group the visible sections one at a time in the Poly Groups subpallet (make sure to hide along the axis with parallel wire frame for clean edges).
Then group split the sections under the Subtool subpallet.
Then close the holes for each slice under the Geometry subpallet (you need to be at the lowest subdivision level or delte lower or higher subdivisions).
As a final step you might want to ReMesh and Project All to conform each new mesh to each old mesh. You can then delete the old meshes if you no longer need them.
Dave