I don't think the LGO will get involved in your case until after the appeal. Given that you seem to have clear evidence of a mistake I would definitely refer the matter to the LGO if your appeal is rejected.
Turning to your questions:
a) You would have to ask to be sure. It is possible they spotted mistakes just before offers went out and decided it was easier to offer additional places than bump 2 children off the list with knock on effects elsewhere. But there may be some other explanation. The LA should know why they went over PAN so it would be worth asking them.
b) Without knowing the reason for the additional 6 children all we can do is speculate. But the point the appeal panel should consider is that the school has admitted these children and, unless they are higher in the admission criteria than your child (they are all looked after Catholic children, for example) or have all been admitted as a result of mistakes, your child should have been one of the additional 6 admitted. To be honest, if they suggest mistakes I would question why they didn't take the cases to appeal as this takes them nearly 15% over PAN. Indeed, I would personally wonder if the "mistakes" were deliberate.
c) Looking at the admission criteria, someone with an exceptional social & medical need would stay within their category but would move to the head of the queue within that category. There is no way this would put someone in category 3 ahead of someone in category 2. Special needs are separate. A child with a statement of SEN naming the school would be admitted automatically. But they would not then be in any admissions category as they don't go through the normal admissions process. And I would be very surprised if there were 6 children entering Reception with statements obtained at the last minute.
d) The net capacity will not include the nursery. The official figure for the number in school should also not include the nursery but it is possible the school has given the number including the nursery. If there are 325 children excluding the nursery it suggests they regularly go over PAN. As they apparently went over PAN by 2 in 2008 and 2011, they must have gone further over PAN in other years. I would ask them for their class arrangement. That will tell you how many children are in each year and how they are split into classes.
e) The net capacity starts with a calculation based on the number and size of rooms. That should exclude the room(s) used by the nursery as that is not part of the school. The calculation produces a range. Net capacity is set somewhere within that range. Unless the nursery stops using a room the number of children in the nursery makes absolutely no difference to the net capacity. PAN should be roughly one seventh of net capacity but they are allowed to set a PAN higher or lower than that indicated by the net capacity.
f) As they are still over PAN there isn't really a space available. However, in the circumstances I would email them saying that you understand there is a place now available and, as you are first on the waiting list, you trust it will be offered to you. If they don't offer you the place you can bring this up at appeal, particularly if it emerges that they don't really have good reasons for the additional 6 children offered places.
Just to add, the rules are that the school needs the LA's permission to admit over PAN. If they have admitted 48 without the LA's agreement they have broken the rules. That is unlikely given that offers are made by the LA. But it does mean the LA should know why the additional offers were made.