Do I have to use Thunderbird to get email notifications?
No, you can use a lot of other email clients like:
- Unity Mail
- Evolution
- Geary
to get email notifications. There some more. The german Ubuntuusers Wiki has a nice list with email clients.
I think you'll like "Unity Mail," which you can install from the Ubuntu Software Center. Per the description, "It displays notifications about incoming mail, shows number of unread messages in the Unity Launcher and shows subjects in the Messaging Menu."
You can set up Firefox extensions as this one so you don't need to use Thunderbird.