Try creating a new profile with an IMAP account - does the problem show for this new profile?
If not, add a POP3 account and copy the correpsponding account files from previous $HOME/.thunderbird/$old-cryptic-profile/Mail/$pop3-account-name.
If this still works (passwords get saved) this could be a way forward.
edit: Another hit when searching for those initial error messages mentioned deleting certain files in the profile directory to let thunderbrid re-create them:
http://forums.mozillazine.org/viewtopic.php?p=14815496
Try exiting Thunderbird and deleting cert9.db, key4.db, pkcs11.txt (new set of files used for security module info) and logins.json (actual passwords) in the profile. I’d also delete cert8.db, key3.db, and secmod.db (old set of security module files) to make certain you have a clean start. When you restart Thunderbird will create replacements for the files it still uses.
Remember to backup the profile(s) before fiddling in case something doesn’t work out.