Ok, here goes. This will be long, so apologies in advance.
As you'll know if you've been reading the other threads, appeals for reception admissions can be very hard to win. Much depends on the admission number (PAN) for the school: if the school admits in multiples of 30, then the infant class size regulations -which limit the class size in KS1 to 30 - will apply and you will only win at appeal if you can show that a serious error has been made which has deprived your child of a place.
You mention class sizes. What are the class sizes at the Catholic school and at the school you've been allocated? You imply that the class size at the Catholic school is less than 30 - is that so? If the class size in reception/KS1 is less than 30 that will leave you with a lot more scope for arguing that your child should be admitted.
What are the admissions criteria for the Catholic school? Do you have the usual letter detailing why you did not get a place? Does it have some places reserved for children who are not Catholic? Or does it place non-Catholic children in a low priority group in its admissions criteria? This will determine where you are on the waiting list, too. Do you know what position you are on the list?
Have you checked that the decision is factually correct, ie they placed you in the right priority group, measured the distance from home to school correctly, etc?
Looking at your grounds for appeal, I am sorry to say that - unless the infant class size regulations do not apply and the appeal panel gives you the benefit of the doubt - they are weak.
Unless your child is sociallly anxious to the point of having a recognised special need, the 'wants to be with his friends' argument will cut no ice. Reception class is all about social interaction and the teacher will be well used to helping children settle in and make friends. Likewise, any competent reception teacher will be able to ascertain which children are on the point of reading and writing and which are at an earlier stage and will provide the necessary support and differentiation. The same goes with the class size argument. Realistically, unless your child's needs are so far outside the normal range (a range of needs that any school should be capable of meeting) that he is already receiving support from a paediatrician/therapist/as appropriate, then the appeal panel is likely to take the view that any school could and should be able to meet his needs.
The last point is about the wetting/soiling. I agree that this is probably your strongest argument but (to be frank) have no idea what an appeal panel would make of it. As far as I know, very many primary schools call parents in to deal with a child who has soiled. The policy of the county school (whatever one might think of it) isn't unusual. In what way is the catholic school's policy different? If you want to make any headway with this argument, you will need a letter from a GP/consultant/other health care professional explaining what the problem is and stating that in their professional opinion (not just in yours) your child's needs would be met at the Catholic school in a way they wouldn't at the county school. If you can highlight differences in their published policies, so much the better.
I am sorry to be so discouraging, but if this is an infant class size appeal, you have to demonstrate that the decision was made in error or was so unreasonable that it should be overturned. None of the social/health/pastoral arguments would come into play. I can understand your disappointment, but nothing you've said so far makes me think you have any realistic chance of winning. BUT things would be more promising if the admission number isn't a multiple of 30, the infant class size regulations don't apply and you can argue that the prejudice (ie disadvantage) to your child of not going to the school is greater than the prejudice to the school of admitting another pupil. You might then find that the panel would be open to persuasion on the social/health/pastoral arguments.
Lastly, do you think there is any possibility that your son does have special needs, as Pheniks suggests? Where a child with special needs is admitted outside the usual admissions round, they can be admitted as an additional pupil, as an exception to the infant class size rule.