this sounds a but fuzzy ni what you want... you can't let users choose to delete the remote emails when it's an automatic process...
anyway, some pointers... you won't want exim, qmail etc.... these are smtp servers, and you've already said you don't want one.
in order to download normal emails from nomal pop3 or imap services, you need fetchmail. while this is a daemonized service, it is not a server, as it doesn't actually serve anything.
and as for the internal imap / pop3, well the standard wu-imap package would suit you, as should others like courier imap.
i still can't see why they can't just connect to the remote servers in the first place though... doesn't sem to be any benefits from your suggestions.