thunderbird meets all your requirements. Not sure what the issue you have with it is.

Anyway, if you use imap, there's no need to backup the mail folders since everything is stored on the server, and yes, you can still do offline email. If you're not using imap on the server, you really have to ask yourself "why". But either way, you can still control where t-bird stores your profile and email folders and back them up if you wish. (though you don't need to since everything is on the server.)

in t-bird, you go into synchronization and storage under 'view settings for this account', click advanced, and you get a pop-up where you can select which accounts and mail folders/etc are kept in sync on your local computer, for searching and also for offline use.

Anyway, good luck finding one that meets your requirements.

-David