Yes, you have to show proof of residency. Sometimes though, you can go to the school board and ask for transfer to another school, but there has to be a really good reason for that request. If it is a psychological reason, if other kids are bullying yours in the other school and damaging their confidence - you may be able to get a doctor or psychologist involved to have your child moved to another school. Please keep in mind, this is my personal view on this questions but may not be correct as the rules for that school board may not allow this - but all you can do is ask.
Good luck with everything!
Yes you will have to show proof of residency. The schools have really cracked down on this in the past years. Problem is, if everyone took their children to schools out of their district, then your going to have over crowding in other schools. That's not really fair to the children, teachers or residents that live in that school district. If your unhappy with the school district that you live in, do something about it. Get involved with the PTA. Try to change the problems and work with the school. If this doesn't work then maybe you need to think about moving to another school district.