How to Start Thunderbird Minimized on Startup

Change the configuration key

mail.db.idle_limit

to

30000000

In Preferences->Advanced->General->Config-editor

Source: http://rainbow.chard.org/2013/02/19/thunderbird-high-cpu/

You can try MailNag:

sudo add-apt-repository -y ppa:pulb/mailnag
sudo apt -y update
sudo apt -y install mailnag-unity-plugin mailnag

Then, run mailnag-config and set up your Accounts (to my knowledge, there's no automated way to import all Thunderbird email accounts), and then make sure the Unity plugin is enabled.

You don't need any add-ons for Thunderbird; in fact, I would actually advise removing the default Messaging Menu and Unity Launcher Integration add-on, because you can open the default email client from the Unity panel via mailnag, and so there's little point for it.

mailnag should automatically set itself as a startup script, so you should get both notifications in the panel (read: the mail icon turns blue) and the overlay style notifications like when you connect/disconnect from a network.

This obviously doesn't actually 'start Thunderbird minimized to system tray on start-up', but it does solve my underlying problem of 'automatically get email notifications in Unity panel on start-up', and the only difference between it and my intended behavior is that thunderbird isn't running. I suppose this is probably why I couldn't find anything about it; now that I know to search for mailnag and found a couple relevant questions.