I suspect what is happening here is that Rhino is converting your spline entities into polylines (PLINES) and squashing everything down into one layer. I believe corel draw is famous (or infamous) for doing the same thing. This is why exploding in AutoCAD helps.
I know a lot of laser/plasma cutter programs will prefer a PLine over an actual spline figure. This enables an approximation of a spline within acceptable tolerances that greatly helps with the speed with which these tools do their jobs.
From the DWG/DXF Export Options on the Rhino website, I'm not able to recognize what the right option may be to change this behavior.
I note there are some Export Schemes. Have you tested using the 2004 or R12 "Natural" scheme? CAM Imperial?
http://docs.mcneel.com/rhino/6/help/en-us/fileio/autocad_dwg_dxf_import_export.htm
It seems that there is a way to create an ACADSchemes and create an AutoCAD export as well.
http://docs.mcneel.com/rhino/6/help/en-us/commands/acadschemes.htm
I'm guessing here, but I bet you would need to select the following curve options:
Rhino Object - AutoCAD Object
Lines - Lines
Arcs - Arcs
Polylines - Polylines
Curves - Splines
Polycurves - Splines
Fusion 360 has a constraint based sketch environment that differs from AutoCAD or Rhino. There is a level of translation and mapping that needs to occur in order for a file to imported from ACAD or Rhino into Fusion 360. You may be able to help this mapping by tweaking the above settings.
I suspect at this time, there are a few work in progress items that are impacting the issue with the splines being editable after importing. I'm still looking into confirming this and will get back to you ASAP.
I hope this helps!
Thanks,
Nathan Chandler
Principal Specialist