How is E value calculated in Slic3r?
To answer your first question:

Your calculations are not wrong, they are correct for a normal layer (uncorrected) layer. These calculations should get you very near the solution. The problem is that there are default modifiers at play that modify the extrusion process which become apparent when you change them or look at the hoover hint in the advanced printer settings section. E.g. see the image below of the "Print Settings" graphical user interface; specifically look at the hoovering hint:



The hoovering hint tells you that there is a 200&nbsp;% modifier at play. What! a default modifier without me knowing? Well...., if we had looked at the 
Slic3r Manual (The Important First Layer) a little better, we read that:


  Fatter extrusion width. The more material touching the bed, the better
  the object will adhere to it, and this can be achieved by increasing
  the extrusion width of the first layer, either by a percentage or a
  fixed amount. Any spaces between the extrusions are adjusted
  accordingly.
  
  A value of approximately 200&nbsp;% is usually recommended, but note that
  the value is calculated from the layer height and so the value should
  only be set if the layer height is the highest possible. For example,
  if the layer height is 0.1&nbsp;mm, and the extrusion width is set to 200&nbsp;%,
  then the actual extruded width will only be 0.2&nbsp;mm, which is smaller
  than the nozzle. This would cause poor flow and lead to a failed
  print. It is therefore highly recommended to combine the high first
  layer height technique recommended above with this one. Setting the
  first layer height to 0.35&nbsp;mm and the first extrusion width to 200&nbsp;%
  would result in a nice fat extrusion 0.65&nbsp;mm wide.


Tada! There we have the modifier from the screenshot; 200&nbsp;% (this is expressed as a percentage over the layer height, and causes that an additional filament scale factor bigger than 1 is at play; the $x$ in your equations).

To answer the second question:

That should be rather straight forward, you know how long the path is and at which speed the head is moving (either at constant speed, decelerating or accelerating) and how much of filament you need to deposit, at the end point all filament needs to be deposited so you can calculate how fast the extrusion needs to be to accomplish that.



If you calculate back from a volume of 8.214 mm2 and solve for unknown $w$ you see that this yields $ w = 0.65\ mm $, and that is exactly what is stated as first layer width in your Slic3r settings; I quote:


  first_layer_extrusion_width = 0.65




P.S. When you look into the source code of Slic3r, if you dig deep, you find that extrusion width is bound by minimum and maximum values, it could well be that that is causing the value to differ from 0.70 mm (200 % of 0.35).