You can set Tbird up to see all the folders in the mail account.
It depends on whether you've set it up as IMAP or POP.
IMAP = the webmail type view(your junk/trash/spam/etc folders. This is where you simply view a remote version of the email on your computer.
POP is where you've downloaded the actual email to your local device(ie. Mac computer).
Same with folders, when you set the account up, it automagically sets up individual directories and sub directories for each account. You can specify at the time of the account where all those messages are to be stored.
(ps. I can't remember if you can reset the location of those folders without affecting the location of any old messages)
I personally prefer to have each individual email account to have it's own directory.
As for backup, I think you can backup some stuff and effectively have a 'full backup' regime but I remember there were some issues with some things not backed up. I haven't used that in years as I use another method.
So I found that the best way to backup Tbird (and Firefox if you use that too) is to use MozBackup. (google it, it's only a small program)
A small independent backup program that completely backs Tbird up into a single (and large) file dated logically.
I use calendars heavily for various things, and have them all synced up with other PC devices and mobile phone.
Problem almost always arises with a new phone(as one example) and my Google calendars usually end up borked to extinction!
I always use MozBackup prior to setting up a new phone because of this, as I can reset the calendars up and tell the phone to use the Tbird calendar first so that it's in sync with Google and then sync with google(even tho it's synced).
Just a note if you use MozBackup for Tbird tho.
When its time to restore a Tbird, you need to have it installed, open it briefly DO NOTHING else, and just close it. Tbird needs to be opened once just to create the default user profile, for MozBackup to have somewhere to backup too.
Once you've restored TBird, its as tho nothing has changed.
As for that last email account. I think it could be the account type or setting to POP.
Have you checked the local account folder for sent and trash for any sent or deleted messages?