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.

This is an honest and heartfelt answer from a History PhD student.

My university was not a tier 1 school, but it is one of the University of California unis. Our Qualifying Examinations were designed such that it is nearly impossible to fail them if you do all the preliminary work prior to the oral evaluations. Essentially, you have to write several long essays that survey the bibliographies of your field and demonstrate your position inside of that academic conversation.

The people who failed their QEs at my school did not put much effort into (or did not complete) the written portion. The oral evaluation is mostly just a demonstration of your memorized knowledge of research/researchers in your field and a defense of your position/dissertation.

I have some bad news, but please take it as best you can: members of your QE committee all have PhD's and they all advise doctoral students for a living. They fully understand the stress and strain of graduate school. They are aware of the attrition rates and even the problem of suicide among grad students (we lost one of our own to suicide during my time in the program).

That being said, there is no way a committee would vote to dismiss someone from the program unless they felt that person was incapable of moving forward with productive research. Unless you are producing research on why vaccines are bad or why Nazism wasn't such a bad ideology, you are likely being dismissed for the reason that the committee feels you are not in league with the expectations of a PhD graduate. Specifically, their decision is not about the fact that they had trouble hearing you or interpreting your answers. It's that your answers revealed to them that the depth or breadth of your knowledge of your topic is not at the level it needs to be at for you to advance to PhD candidacy.

There is another factor in all this you might not be aware of: At the time of QEs, your advisor submits a holistic evaluation of your progress to the committee and to the administrative department with his/her recommendation about your candidacy. It is not at all unheard of for advisors to act with favoritism or to mistreat advisees, but the more likely explanation is that your advisor feels the same way the committee does.

If you are going to be dismissed, you are owed an itemized list of reasons why. You could try to counter-argue those points. Your strongest argument might be that your advisor is guilty of dereliction of responsibilities to you, but that is a hard thing to prove against a tenured and respected faculty member. Even if you do win, he's going to be even more obstinate moving forward.

My honest advice is this: you do not really have a case. Arguing that you studied really hard and used scientific rationale to the best of your ability during your exam is insufficient; it's the kind of thing undergraduates tell their TAs all the time when they fail a test.

To address your first question: "My question is how I can a valid arguments supported by evidences from my counselor & physician on the ground of mental health issues and ADHD?"

You cannot make this argument. You are literally saying "I should be allowed to remain in this program because I have mental health issues that prevent me from performing acceptably in this program."

To answer your second question: "The other question: what is my chance of admission if I apply in the future for another PhD program?"

If you wait a few years and have considerable success managing your mental health issues, I see no reason why you couldn't get into another program - but you'd have to be forthcoming about the fact that you were removed from your first program, and you'd have to demonstrate that you are now capable of succeeding. That will be tough.

Two other things you should consider: Graduate school and mental illness are an explosive mixture. Grad school is decidedly not good for anyone's mental health, and if you already have these issues, I can't imagine it's a good fit for you. Also, there is very little actual use for a PhD unless you want to teach at the university level. An M.S. is plenty to land you a decent job outside of academia and sometimes still inside it as a researcher. Take your Master's (which you were probably awarded during your second year) and grab a job! There is no shame in not finishing the PhD.

edit: for clarification, I do not mean to imply that a person struggling with mental health issues should never be in graduate school. But I do mean that it is reasonable to expect that grad school will exacerbate those issues. It's a big enough problem that this warning is repeated to virtually all incoming PhD students, and it's a topic that's discussed openly during the program. Good schools will have good resources available to students who feel mentally unwell during their program. But if you were my son/daughter and you struggled with those issues, I would say "Don't go."