Gaba, I'm a panelist so can answer your questions 
I'm allowed to claim for actual travel costs if traveling by public transport, on production of a receipt for those costs. Alternatively, I'm allowed to claim mileage at something like 55p per mile, but that is subject to tax, or 40p per mile untaxed. Mileage claims are subject to audit. I'm also allowed to claim cost of parking if I drive in.
The regulations require the Local Authority to advertise every 3 years. I applied simply by googling my LA and 'independent appeals panel' then found a form and sent it off.
The Manager of the Independent Appeals Process is an LA employee, but is at 'arms length' of the LA and acts independently. If parents phone him he will advise them as far as is legal, as to the process and what their role in the process is.
To be on the panel you must satisfy the co-ordinator of the service that you are not an 'excluded person'. The list of exclusions is in paragraph 1.7 of the appeals code :
"a)a member of the local authority which is the admission authority or in whose area the school in question is located;
b) a member or former member of the governing body of the school in question;
c) employed by the local authority or governing body of the school in question, other than as a teacher or teaching assistant4;
d) any person who has, or at any time has had, any connection with the authority, school or any person in sub-paragraph c) above which might reasonably be taken to raise doubts about that person’s ability to act impartially;
e) any person who has not attended training required by the admission authority arranging the appeal panel."
In my Local Authority, the applicants fill in a form, where they declare any connection with any school in the Local Authority, past or present, plus any connection that their direct family hold.
I am excluded from hearing appeals for My DD2's old school and DD2/3's current school, and it's unlikely I'd be asked to hear an appeal for my DH's school, because he is a caretaker there, although I could do. My DD1's school is a special school, so they don't fall under these arangements.
The service has a pool of panel members. They try to balance the appeal hearings carefully. e.g. if the panel are all male, then they have a female clerk. The majority of panel members are of the older generation and male, as that's the demographic of the applicants. I am a female in my 30's, so I am likely to be brought in to balance the panel on occasion.
There has to be at least one 'lay member' and one 'non-lay member' per panel - that is at least one who has experience of education (other than their own!) and one who doesn't have any connection with education.
The appellants will always know the names of the panelists prior to the hearing, and whether they are 'lay member' or 'non-lay member'.
If you were to find out something disturbing, you should speak to the appeals service manager to alert them. It depends what it is really?
In terms of your analogy, the panel have nothing to do with the admissions department. The expenses are laid out in the Code, and are non-variable. Equally, the existence of the panel and their function is laid out in the Code and is non-negotiable. There is no incentive in any way to find in favour of the LA. In fact, we have great power to over-rule the LA on our discretion in prejudice cases. Our hands are quite firmly tied in Infant Class Size cases though.
If there were 55 appeals last year and not one succeeded, is it worth appealing?
It's always worth appealing because it is your right in law and you may be successful.
If there were 55 Infant Class Size appeals, then it's unsurprising that they weren't successful. They very rarely are (about 0.4%), because unless a mistake has been made, the bar is set so very high.
If there were 55 prejudice cases and none successful, it depends what the conditions of the school in question are. A very oversubscribed, bursting at the seams, high EAL, high SEN, with little communal space would be far harder to win an appeal for, than a school with huge communal spaces, 30 in a KS2 class and very little SEN/EAL, generally undersubscribed but happen to be full in the applicable year group. This would be because the prejudice would be far greater to begin with at the first school, so the case would have to be much stronger for the appellant.