I am trying to understand if there is any relation between solid model position under Fusion360 and its position when the solid is dropped into Simplify3D.
So far, I cannot see what the initial orientation of the dropped into S3D solid bodies is determined by.
Open your model in Fusion and at the upper right corner select "bottom" on the viewing cube and that will show you and be the surface that is placed on the bed in the slicer.
If you want a different surface to go to the bed in the slicer then you will need to change the orientation in Fusion.
Have not used fusion yet, but I assume you have 3 planes to start with. Take note of which plane you model is on , export it and bring it into S3D. If it is not in the same orientation, flip back to F360 and figure out which plane it would have needed to be on to be right side up. Next time use that plane so the model will land correctly.
If you don't already know, with the model on the bed, hit Control-L then click on the surface you want on the bed and it will flip.
In Onshape the top plane is where i start all of my drawings so the end up right side up.