What is the chance of making a successful appeal to dismissal decision from a PhD program after failing the qualifying exam in the 2nd attempt?
I'm sorry for the tough situation you find yourself in.

First things first: no one outside your university and department can speak with any authority whatsoever as to what this particular group of faculty administrators will do.  At best they can speak in general terms and give general advice.  That is what I will try to do.  Let me also begin by indicating my own experience: I am a mathematics professor at the University of Georgia.  For the last three academic years I have been the Graduate Coordinator of the mathematics department.  Thus if you were in my department you would be appealing to me (more precisely, to the committee of which I am the chair).  

It seems to me that you are getting overly caught up on the legalistic aspects of your "appeal."  You grasp at several things that might perhaps lead to bias on the part of committee members.  But let me be honest: in the kind of academic culture of which I am familiar, this is not a very good strategy.  Without evidence of clear and bias that changed the result of the exam, I think it is very unlikely that (say) the director of the PhD program will do more than inquire of the faculty members whether there were any serious problems or irregularities with your exam.  When they say "No," there is not really any other group that can spring into action at this point, again unless you have something much more serious to allege and more material reasons to allege it.  

Here is the reality of the situation: the committee who administers a graduate qualifying exam has extremely broad latitude in deciding whether the student passes or fails.  There is probably not going to be a "rubric" that they did or did not follow or anything of the sort -- rather, they are empowered to use their best judgment.  To be honest about it, a student at this stage is probably not in a good intellectual position to question their decision: you don't have the experience to know whether their standard is a good one.  

Rather than thinking of the appeal as an argument that you have to win, I suggest that you appeal to the sympathies of the most sympathetic faculty member in sight.  In particular, I do suggest that you mention your health issues of anxiety, depression and ADHD.  As a longtime faculty member and administrator, I can tell you that (i) many students have these problems, (ii) they can become significant impediments to otherwise very strong students and (iii) these issues are (happily!) not nearly as stigmatized as they used to be, to the extent that "zero sympathy" for such a student is not such a tenable position for e.g. a public university in the United States to take.  If you bring medical documentation of the problems, explain the serious steps you are taking to treat the problems and ask for another chance when your health issues are under better control, then I think you have at least a fair shot at getting another chance.

Good luck.