How to extrude the faces normal to them and then once normal to one side, and then in the opposite direction to make something like on the picture but following the seat edge? I thought about Shrink/fatten but it does not work.
You can easily CtrlLMB to extrude towards the mouse. (if you are talking about making the curved back)
If you are just trying to make the edge rounded (a Subsurf modifier will help will the smooth roundness) add an edge loop on the out side edge. The pink line in the image below is the new edge loop.
Then switch the Transform Orientation to Normal and move the new edge along the normal Z.