Firstly I would say that you have no ability to question the decisions made in terms of your child's health issues or the issue over your work arrangements and the LGO will not be interested in such arguements.
The changing PAN makes me ask you whether this is a welsh school or an english school as the rules are lightly different in wales and could lead to this changing Admission Number. How do you know that the PAN was changing from one year to another? Did the admission authority volunteer this information?
There are potentially questions to be asked about the PAN depending on how you answer the above questions. What should happen in england assuming that the PAN is 57 is that the panel will hear the arguement by the admission authority to not admit any further pupils. They have at the end of stage 1 a number of decisions to make but the key one from your point of view is whether the admission authority have made their case and that the school cannot admit all 18 appealants. As the panel will note that infant class size regs comes into effect at 60 I would say that it is a certainty that the admission authority will make their case.
The other key question is whether the admission arrangements were correctly administered. This should be stated on the letter you got confirming that the appeal was unsuccessful and I would assume at present that the arrangements were not at fault.
At stage 2 in the appeals the panel hears all the personal circumstances and has a decision to make as to how many of those cases excede the prejudice to the school. That is something only the panel will know but the panel must not allow the ICS regs to be breached so the maximum they can admit is 3 to get to 60. The panel has to weigh up the cases and decide which three have the highest level of prejudice and admit those three.
I therefore do not see any obvious faults with the methodology that the panel will have followed to get to admitting the maximum of 3 at appeal, given that the panel can only go on the strength of the cases presented and not where you might or might not be on the waiting list.
That brings us back to the question of the official Admission Number, is it 57 or 60? If it is actually 60 then I would say you do have a case, because given the assumption that your neighbour got a place you would be the next on the list for a place of which there were 3, assuming this on distance only. But I suspect that the admission authority would not make such a basic mistake. The key is what is the quoted admission number in the LA book on primary admissions. If it says 57 then you have no case, if it says 60 then you have a case.
I would agree with PRH that they should not be using an OS map to decide distance. However many LAs do use a computerised OS map system for the measurements and this is very accurate. Each building has what is called a seed point on it and this is the point that it is measured from, this would accord with the comment about the centre of the house as this is usually where the seed point is - it will potentially not take into consideration any extension. I don't believe that the LGO will be interested if this point has been used but you then say about it being measured from the furthest corner of the house which is clearly not correct. Did you raise this at the appeal? If so then talk to the LGO.
What was the distance to your house and the distance to your neighbour who got a place? They should be very close, if not then they either did not get the place solely on distance or there is a mistake.
Sorry this gets very complicated and you do need to know all the info.