No, if I understand your question correctly, you do not start renumbering with each chapter. Think about how confusing that would be to keep straight if the pages got out of order. The pagination is strictly for the editor or agent to be able to keep the pages straight so you start with page 1 and go through to the end.
Georganna, I have never seen anything that allows changing the pagination in the header. Where would that be? No, you don’t have to put the header on page 1 but it will still consider that as page 1 and number the next page as page 2 as far as I can tell. If there is any way to force it to force it to number the second page as page 1, I’ve never found it. But I’d be happy to be proven wrong if somene can tell me some way of doing that.