Ok.
Starting with DS1 - you need to appeal for the school you want (the academy) not against the schools you don't want. So the school has marvellous facilities that you really want your DS to access (preferably some that the other schools don't have), they are full to PAN but not up to the Infant Class Size limit, so you need to ideally show that the school could accomodate another Y2 child without major problems - look at the room size of the Y2 class, the overall school site - how are the communal spaces, such as the hall, etc.
Have you been offered an alternative school for your DS1? How far away is it?
Your DS 2 is going to be far more difficult. The appeals panel can only consider the appeal for the year group he has applied for. That year group is not only 7 over PAN, but 3 over the Infant Class Size Regulations (incidentally, do you have details of why those 3 were admitted?). That means that for another child to be admitted within the Regulations, but without a Statement, 4 children need to leave that class/year group.
You are very unlikely to win at appeal, unless a mistake has been made.
The medical information can't override the ICS rules. The only way of circumnavigating them would be for the LA to issue a Statement and name the academy. Have you thought of applying for a Statement - does your DS currently get significant help in school?
The appeals panel can't consider an appeal for your DS to join the Reception year. Presumably, also, the school doesn't actually have a Reception place, because otherwise they could admit him as an out-of-year pupil in collaboration with the Local Authority? Reception year also has the Infant Class Size regulations, so that won't help you.
Sorry not to be more positive. I've been told that for our County, the success rate of ICS appeals is something like 0.4%.