These two issues are probably not related at all.
Unity Tweak Tool
When Ubuntu is upgraded, 3rd party PPAs (repositories) are disabled. Maybe you're still using an old Unity Tweek Tool that is NOT compatible with Ubuntu 13.04.
Check your Unity Tweak Tool version. Open a terminal and paste this command:
dpkg -l | grep unity-tweak-tool
The latest version available at the moment of writing this is
0.0.5-0~136~raring1. If you have an older version, go to the official PPA, add it again to your system and then update.
You are not providing too much information. This can be caused by several problems. These are some of the most common:
Check that there is disk space available. You can do that on several ways, but a very simple and quick one is by using the terminal. Launch a terminal from Unity, and run this command
You'll see one line for every mount point. If you see some mount point with a large usage (>95%) try to free some space.
Package manager Status
The package manager status could be bad, so some packages can be mid-installed, or mid-upgraded. You can check that by issuing this command:
sudo apt-get -f install
If that says 0 packages for updating and such, everything is fine. Don't worry about not updated packages. You can have some not updated packages and still everything is fine.
Maybe your Thunderbird user profile is corrupted somehow. The easiest way to test this is to create another user, login as that user, and try to run Thunderbird.
If Thunderbird is working with that user, then just simply move your old Thunderbird profile to some other location.
mv ~/.thunderbird ~/.thunderbird-old
And then try to start Thunderbird again. You will lose your configurations, though. But worry not, you can still configure it again to access your local mail, if available.