Here is my take on your last questions but prh47 is clearly the expert (and no, I don't think she sounds tough- just sticking to the rules as a panel should):
The whole admissions/appeal process is based on the assumption that every child in the school will be disadvantaged if the school takes on more than its designated number of pupils. It follows that the only sensible reason for doing so anyway is if the disadvantage arising to one particular child is so great that it outweighs this disadvantage caused to all the children. (or of course if the LEA has made a mistake).
Most panels would not think that going to a school with slightly less music provision would be such a huge disadvantage for a child- parents can always provide music tuition outside of school and indeed that is what most parents, even of very gifted children, have to do.
"Ok I am sounding desperate here but the distance measured for the preferred school was as the crow flies. I found out today that someone I know was accepted at the school who is closer than we are measured by as the crow flies but actually has a longer journey than us, is this fair and would that be taken into account?"
By now you should have the list of admissions criteria- the LEA will adhere to those. So if the criterion is "as the crow flies", then they can't change that in the case of one child to be "longer de facto journey"- if they did they'd have re-do the whole admissions process to make sure they were doing the same for everybody. The criteria that are published are the ones they will stick to.
(In the case of our school, this led to rather bizarre results as children who lived in the same road as the school (out of catchment) were rejected while children who lived at the other end of the parish were admitted because they were in catchment. Parents complained but were told that they cannot change the criteria during the admissions process).
"Would they not have to offer me a local school or are they within the law to offer me any available school even if it is too far?"
That depends on the definition of too far. They have to offer a chance of education= a school that your dc can actually get to, but if it is too far to walk, they may offer transport.
"In order to avoid the preferred school spending extra money for a teacher, would placing my child on top of its reserve list be a possibility regarding the circumstances of the offered school, the stressed that it has all caused and that there was no other school with a place at the time that I enquired?"
Only problem I can see here is that all the other children on the reserve have equally been rejected, are equally faced with the choice of less satisfactory schools and can be expected to be equally stressed.
"As I mentioned in a previous message we need the after school clubs that the preferred school offers but also the help of a friend whose child goes to this school who would help with childcare for free. Unfortunately we cannot afford to pay for a paid carer to do the school run and after school care."
Again, lots of parents could argue this, lots of parents are struggling financially, lots of mothers cannot afford to go back to work because of the cost of childcare: it would be hard to prove that similar problems did not equally affect all the children on the reserve list. In fact, most people do not have the luxury of having friends who can do free child care, so it would seem unreasonable to the LEA to give your child priority because you are so lucky.
Agree with prh47 that the threshold is very high for proving admissions mistakes. The panel did not accept that the LEA had made a mistake when they refused to place our wheelchair bound dd into the special considerations category and allocated us a non-wheelchair accessible school: their reasoning was that we should have provided more medical evidence than just a letter from the GP. We still got in on special considerations, because this was secondary, but wouldn't have done if we had had to rely on the mistakes argument.