Starting from the end of your post:
The panel will usually be made up of 3 people. At least one must be a 'lay-person', that is, a person with no experience of having been employed in the education sector, and at least one must be a non-lay person, which means that they have been employed in the education sector. The panel will also have a chair person. Nobody on the panel will have any connection to the school or the Local Authority, so they can be impartial.
does mentioning that I am a SAHM with willingness and time to help with trips/reading with kids/PTA make any difference?
No, not at all. This is about your child, not about your ability to help the school. Also, you'd presumably be willing to help wherever they went to school? Taking it wider, a panel couldn't not award a place because a child didn't have a willing parent to help a school, so it just can't be a consideration.
The reasons you give are all nice ones.
The key with the class sizes is to see how big the classes are as a whole, because you may be able to argue that although they are nominally full for year 4, there is space to accommodate your child in the class, due to underfilling in year 3, for example.
If they run mixed year classes, you'll also have to look on to whether year 2 and year 5 are a big year group, and what impact an extra child would have when she becomes year 5 and the year 5s become year 6, because of something called 'future prejudice'. So if year 4 is already a big year group, and year 5 is, say, 3 over number already, and year 2 are 2 over number, they could say that next year, when they become year 5/6, there will be too many to safely accommodate. The risk of that is slightly lower in this case, I would think, because the year group is split over 3 different years (yr3/4, yr4, yr4/5) so they should still have a degree of flex.
Generally speaking, it will come down to how big the school is, how oversubscribed it is already, and how much of a fight the LA is willing to put up. The harder they argue, the stronger your case has to be. I have been in appeals where a HT has said "Oh no, we'd love to take another child, the more the merrier..." - the LA rep despairs, but job done.