Awful styling in MMA10's Plot

Try to add the option: PlotTheme->"Classic"which transforms the plot to the style of MMA7:

 pad = 100;
Fig8 = Plot[{2 Cos[x], Sin[x]}, {x, 0, 1}, PlotTheme -> "Classic", 
  Frame -> True, PlotStyle -> {{Thick, Black}, {Thick, Dashed, Gray}},
   AspectRatio -> 0.6, PlotRange -> All, 
  FrameLabel -> {{"Froeschlé HE", "Chirikov mLE"}, {"Chirikov mLE", 
     "Froeschlé mLE"}}, 
  PlotLegends -> 
   Placed[LineLegend[{"Chirikov map", "2D Froeschlé map"}, 
     LegendLabel -> Placed["p(x)", Above], LegendFunction -> "Frame", 
     LegendMargins -> 1, LabelStyle -> 5], {0.3, 0.55}], 
  BaseStyle -> {FontFamily -> "Times"}, 
  FrameTicks -> {{{{0, "0.0"}, {0.4, "0.4"}, {0.8, "0.8"}, {1.2, 
       "1.2"}, {1.6, "1.6"}}, {{0, "0.0"}, {0.4, "0.2"}, {0.8, 
       "0.4"}, {1.2, "0.6"}, {1.6, "0.8"}, {2, "1"}}}, {{{0, 
       "0.0"}, {0.4, "0.4"}, {0.8, "0.8"}, {1.0, "1.0"}}, {{0, 
       "0.0"}, {0.4, "0.2"}, {0.8, "0.4"}, {1.2, "0.6"}}}}, 
  ImagePadding -> {{pad, pad}, {All, All}}]

The result should look like the following:

If this is not what you are looking for, please explain more precisely.

Have fun!

On some computer systems, the OS & Mathematica seem not handle fonts properly when exporting to EPS. I am not able to check all systems, but it is possible that converting an expression to PDF first and then to EPS might work. Importing the PDF shows that the font glyphs in the image you get have been converted to Mathematica graphics primitives (FilledCurve usually). Exporting the graphics to EPS should work on all systems. The only possible hitch is whether the fonts were rendered properly in the PDF conversion and import.

Code:

Export["Fig_8.eps", First@ImportString[ExportString[Fig8, "PDF"]]]

General-use function:

exportViaPDF[file_String, expr_, opts___?OptionQ] := 
 Export[file, 
  First@ImportString[
    ExportString[expr, "PDF", "PDFOptions" /. {opts} /. "PDFOptions" -> {}]],
  opts]

It's hard to set defaults for options to Export. See Pass Options to Export[]. On the other Export is very forgiving of options that make no sense. It seems to ignore them.

Side note: In some random testing, I found the default PDF option setting "AllowRasterization" -> Automatic leads to error messages when the expr is 3D graphics (V10.0.1, Mac OSX 10.10.1):

Import::general: Unsupported Shading type 7 >>

But both the True and False settings worked fine. Example:

exportViaPDF["foo.eps", Plot3D[Sin[x y], {x, -2, 2}, {y, -2, 2}],
  "PDFOptions" -> {"AllowRasterization" -> True}, ImageSize -> 100]