My apologies, I reread your post and left a few items out.

Regarding the appeal letter, unless you feel you were really mistreated during the examination, try to avoid bringing examiner conduct into question as a possibility for having problems.  It will only make more problems for you.  Remember, he/she is already in the "club."

As far as polygraph/background investigation order, it is true that the background did not start before a successful polygraph before.  However, an FBI contact has indicated that the policy has changed and the BI MAY begin before a polygraph.  Unfortunately, the BI will come to a screeching halt should you fail the polygraph.  That would explain your in-progress BI.

I think your "interesting and mind stimulating work (not to mention important to our country)" comment is a bit of a stretch.  Many agencies do not use the polygraph examination.  Goverment work is not the only work that is interesting, stimulating or important to our country.  Think of it this way - if the job exists, the work is necessary and someone must do it.  That makes it important.  The rest is up to you - any job is really what you make it.  Check out other agencies and don't forget the private sector while you're looking.  Good luck.