Is there a way to send data/messages from the chrome extension to the thunderbird add-on? I didn't find any API.
Use Native Messaging to send all information required to compose an e-mail (to, subject, body, attachment) from your Chrome extension to a native application, such as a Python script:
https://developer.chrome.com/docs/apps/nativeMessaging/
The Python script (which you must program yourself) then calls Thunderbird from the command line, with the appropriate command-line arguments:
http://kb.mozillazine.org/Command_line_arguments_-_Thunderbird