Why do they ask these questions on the application form
The short answer is because they are required to do so by the Admissions Code. The longer answer is that very occasionally someone will put something that shows they qualify to go into a higher admission category. An example would be mentioning a medical condition that means they need a place at this school. But most of the stuff parents put in that box is just ignored. They can't make subjective judgements on your reasons for wanting a school. They have to judge your application purely by the admission criteria.
Is it because we are applying late do these things not matter
No, they wouldn't generally matter even in the normal admission round for the reasons I've given above. But, as this is a late application, all that really matters is whether or not the school has a place available. Which admission category you are in, distance from school, etc. will only be used to determine your position on the waiting list.
The PAN is 60
That means any appeal will be infant class size so I'm afraid your chances of success are very low. You can give it a try but you need to be realistic about your chances.
I know they sometimes do go over this as they have done before
There are certain specific situations where they can go over the number. For example, if a pupil misses out on a place in the initial allocations due to a mistake they should be admitted even though the school is full. Similarly if a child gets a statement of SEN naming the school they must be admitted. There are also a number of other situations where this is allowed.
Who decides on the grounds to appeal and what type of appeal it is?
The type of appeal is determined by the size of classes in infants. If any of the classes would have 30 pupils with a single teacher if all three years were full the appeal is an infant class size case. The rules as to what grounds can be successful in an ICS case are set by the government (and haven't changed since the class size limit was introduced).
When it is not an ICS appeal you can use any grounds you want.