So, I'm looking for something like this: One program I can run on one of my servers, which has all the passwords and can see the inboxes of & send mail as any of my 3 addresses, which I can then connect to on all my devices either though a mail client or a webmail thing (4/5)

Not quite, actually, since Thunderbird has no way of syncing settings between multiple devices, so whenever I set up a new device (or re-set-up an old one) I have to type all my credentials in again, and when one of my passwords gets reset I have to type it back in on all my devices. (3/5)

Usually this would be a pain since I'd have to use 3 different webmail pages, but my mail client, Thunderbird, supports multiple mail accounts, so I can have all of them in the same program. Great, right? (2/5)