B4r4joon's case was essentially that her child should have been given priority on medical grounds but the LA failed to do so.
Your case for the first school is that you submitted the supplementary forms but they lost them. That is unlikely to succeed, I'm afraid, but it seems to be the best you've got at the moment.
Your case for the second school appears to be that they have broken the Admissions Code by asking a prohibited question and that you believe the answer to this question was being used to illegally give priority to some children.
If the paperwork from the second school is as basic as you say that is a serious problem. They are required to say why admitting your child would cause prejudice and make it clear whether or not they think it is an infant class size case. They are required to state the school's net capacity. I would also expect them to give the class arrangement and the number of pupils in each year. If you lose your appeal for this school I would recommend a referral to the Local Government Ombudsman on the grounds that the school failed to comply with the Admission Appeals Code paragraph 2.20 and this hampered your preparation for the hearing.
The fact the second school think they can get away with such a poor case makes me wonder if they've had the same appeal panelists for a long time, to the point where they are no longer truly independent. Watch out for anything that happens in the hearing which gives the impression that the panel isn't independent.
The important thing, though, is to give it your best shot and hope the panel come down on your side.