I have always used local storage and local backups for emails, contacts, etc. Of course I could have a drive go bad and I would only have myself to blame if I didn't have a backup.
Once you set up Tbird to use POP (as opposed to IMAP), all emails pulled from the POP server are removed from that server and only exist locally on your PC. There may be some delay before they are actually deleted from the server. There is an option in Tbird account settings to "leave mail on server". This is handy when you have two devices for fetching mail but don't want the secondary device to delete any.
Tbird will keep your emails indefinitely except maybe the trash or spam folder. Once you start storing your emails in Tbird, you will want to use Tools->Export at least once a year to create a backup.
A few newbie questions, please..
I don't use IMAP, so don't know the deletion policy. I also have not played with the yahoo email enough to be sure of how it will behave.
Typically you can set up any number of devices with POP settings, but the first device to read an email might remove that email from the server. If so other devices that later check your inbox will not see that email. If that happens and you don't want it to, there is an email setting in Tbird or your phone to "leave email on server", in which case you get a copy of the email but it also remains on the server so your other devices can read it as well.
