It can be helpful to know which schools we are talking about as it is then sometimes possible to give more specific advice. It is up to you whether you want to post that information in a public forum. If you don't want to do that, you can send a private message to myself and/or Admission - click on "Message poster" to the right of our names in the messages we have posted.
LA = Local Authority, in other words the local council that covers your area and has responsibility for education. Frequently also referred to as LEA (Local Education Authority).
Meeting with head teachers won't directly get you a place. The system doesn't work like that. However, it can be a good thing to talk to the head for all kinds of reasons, not least of which is that you may pick up information which will help you at appeal.
If there are places for your older boys at the CofE school I would go for it. Provided they give sibling priority that will mean your youngest will be at or near the head of the waiting list, increasing your chances of getting a place for him. Whether or not he would end up with a place would then depend on how many children leave.
The letter from the vicar and the letter from the clinical psychologist will help but it is impossible to say how much. It will be up to the appeal panel to decide how much weight to give that evidence.
I would be careful about placing too much reliance on disparities between what schools say on their websites. It may be that the CofE schools really are much stronger on religious education but equally you may find that there is very little practical difference.
The factors you mention (music, sports, etc.) are good and you should definitely bring those up in your appeal.
I would recommend accepting the offered place. It gives you a fallback in case your appeal fails. It also avoids giving the impression that you are trying to force the appeal panel to admit your son - unfortunately all panels have come across parents who believe that the appeal panel must admit their child if they reject the offered school. If the panel think you are trying to take that approach, effectively blackmailing them, you are less likely to get the benefit of any doubt.
When appealing for a CofE school a letter of support from your religious connections may be helpful. Equally the appeal panel may decide it is irrelevant. It can't do any harm to put such a letter in your evidence. I can't immediately think of anything else that would be helpful in this respect.
On the distance question, you may not be aware that you are entitled to free transport to and from the school for your sons given how far it is from your home. That may well take the form of a taxi picking them up in the morning and dropping them off in the evening. The panel will be aware that you are entitled to free transport. I would therefore avoid getting into the details of how your sons would get to and from school and concentrate on the potential damage to their social development. Having said that, it may be worth finding out what the LA will provide.
The admissions number for the CofE schools should be in the council's admissions booklet which you will find on their website. If you can't find it the council will be able to give you this information. They are required to answer any questions you reasonably ask to help you prepare for your appeal.