Hi BruceBanner
I hope other Forum members will jump in with their understanding as well. Here are a few thoughts
1) Many have had trouble with FastStone for color management so I will put up a yellow flag. I have avoided it for that reason. There are "supposedly" settings in FastStone to turn on color management and also I have seen posts that indicate you need a little adder program to have good color management with FastStone. You would have to ask someone other than myself as I just avoided it given the color management issues in the past.
2) I would not trust the Export As preview or the Save for Web preview as the resource for soft proofing. I have not seen detailed explanations of how those previews within PS work (or don't). I see both of those functions as "helper" programs to prepare for a particular environment with options on converting to sRGB (if not already done), possible output to different sizes, embedding profile or not and in the Save for Web for fine tune control of compression settings and many other parameters. I do not see those functions as an accurate preview for what your are going to get in the final output (though they should be reasonably close)
3) For a more accurate preview in PS I suggest two approaches
a) Use soft proofing in PS which means understanding the options available such as rendering intent, black point compensation, etc and then you have a more accurate representation of you image as full screen
b) Save your original file with all its Layers etc as PSD or Layered TIFF in the original color space and create a second version for outputing using the Edit > Convert to Profile command. Again, using this command you should know all the underlying options for rendering intent black point compensation, and of course targeting you final color space desired.
4) In Lightroom you have soft proofing and also, if you import to Lightroom with an image that already has a color profile embedded, it will preserve the profile for viewing purposes. Note, for the most accurate representation, view in the Develop module.
If you follow the above approaches, let us know if you have any remaining discrepancies
Hope this helps
John Wheeler
PS - Welcome to the wonderful crazy world of color management.