Yes. IMAP is just like you described as wanting. I use it at work. In Mozilla, I get the headers when I select get new mail. They remain on the server until deleted and don't download until selected.
Even after downloading, you can't read them again unless you have connectivity to the mail server. i.e. continually re-sync with the mail server.
There is an offline option to download all messages locally for offline use, but haven't used it. IMAP is so much better than POP cause you can just delete what you don't care about and not have to download huge files to delete them (although, in mozilla with POP you can tell it not to download messages over a certain size). Of course, the mail server has to be an IMAP server. Simply trying to use IMAP on a POP3 server won't work.
Hoped this helped, seemed to be all over the place.