What exactly happens at the AC - I honestly do not know. I believe however, that a group of judges, assisted by clerks, review the recording and the records. they are not looking to see if they agree with the ALJ. Rather they are looking to see if there is sufficient support for the outcome reached by the ALJ, and to make sure the ALJ followed all the rules and procedures to assure you received a fair hearing.
I would not expect nor rely on the Appeals Council combing the case for errors. It is pretty clear that you need to identify the errors from the ALJ hearing and raise those to the Appeals Council (although admittedly the Appeals Council is far less of a true appellate review than federal/state appellate courts). Having said that, I would be surprised if both the ALJ hearing the case on remand and your representative did not address the issue that led the Appeals Council to remand the claim in the first place. Nevertheless if this is the second time around at the Appeals Council for you on the same claim, then I doubt they will view the case in a vacuum and not recognize the issues that originally supported a remand.
It is impossible to give specific answers to questions without meeting and fully discussing all of the potential issues that may not be addressed by your question. The answer(s) provided should be considered general information and are not legal advice. Viewing this information is not intended to create, and does not constitute, an attorney-client relationship. The answer provided is intended to educate you and point to issues for you to raise in a consultation with a lawyer. You should not take any action that might affect your claim(s) without first seeking the professional opinion of a licensed attorney.
Our practice has had some experience with the AC and it's been fairly negative, to be honest. For example, one recent case's opinion had numerous errors in it. Evidence was cited that didn't exist, there were inconsistencies between statements of the facts, and it was pretty clear the judge or clerk who wrote the opinion did a sloppy job. In the end the ALJ's decision was not overturned.
The point is that regardless of what happens behind the closed doors of the AC, don't get your hopes up.
For specifics on what happens when your case is being reviewed by the Appeals Council, you may go online at ssa.gov rules and regulations and look under HALLEX. The better answer may be to have a meeting with your current representative to address your concerns. Since the ALJ must address the order of the appeals council, his/her failure to do so could be a basis of having your case remanded for a second time.
I would like to address what I believe as a social security disability attorney of 20+ experience is the critical issue, rather than your laundry list of what would otherwise be very valid questions. The critical issue is the lack of confidence you have in your "new representative". By inference you have had at least 2 attorney/representatives that you are unhappy with. This fact creates, for me, three possibilities: (i) you have been unlucky/unwise in your choice of representatives, (ii) you have a weak claim for disability and you are taking it out on your representative(s), or (iii) you have had hearings before an ALJ who tends to deny claims and you are taking it out on your representative(s). You will have to look at the situation, and yourself, and clearly determine which is correct. As a social security attorney of many years I believe the level of quality for disability attorneys is quite high, so possibility (i) is IMO the least likely. Good people with good claims get turned down: it doesn't mean it is the representative's fault.
Your moving forward with an appeal, with a representative you claim is not addressing the issues, is a very bad situation. I am more concerned about that than anything else you bring up.