Completly beveling edges on model I am trying to get rid of all corners on upper part of my object, so it would be nice completely circular (blue line) while maintaining lowest possible number of vertices. Now there are many visible edges on the upper part, but I am trying to get rid of them. Also this upper part has many visible edges if I try to bevel parts of it even more. (image 2)


Tried with subdivision surface, but I get way to many faces
Tried playing with edge split and smooth operator
Tried beveling each edge, but then I get even worse result than on bottom image after cleaning a little.


Would really appreciate any advices on the most simple way to achieve this.

File:
If you do triangulate (ctrl+T) you can see why you have problems with your mesh structure (you have some Poylgon-faces there which will be exposed by this)

I would strongly suggest to do some manual retopology to clean the mesh base structure. It is really not that hard as it sounds. 

The easiest way to do this would be to just select some points, copy and seperate them from that object and than connect them to make nice quad-faces, then you could use edge slide (press "g" twice) to make the positioning of the vertices more even if you need to... etc.:



So you can see how the geometry affects the subdivusion surface behavior here is a comparison: (your faces above, quick manual retopology below) 



If you want to maximally reduce the real geometry you can apply the modifier, select the loop cut with Alt-select and delete it with "dissolve edges":