2020 Update:
If you are writing a MailExtension (this is how WebExtension are called for Thunderbird) you should have a look at general debugging in WebExtensions, e.g. https://extensionworkshop.com/documentation/develop/debugging/
You should probably also take a look at the general developer documentation on MailExtension at https://developer.thunderbird.net/add-ons/about-add-ons
The thinks listed below there written for the Legacy Overlay Extension, which do not work in newer Thunderbird versions.
logging messages
As described in https://developer.mozilla.org/en-US/docs/Debugging_JavaScript, there exist 3 different consoles in Gecko. The one best accessible in Thunderbird is the Error Console. You can write messages to it through the nsIConsoleService. If you don't mind that the log messages show up as an error, you can also simply use Components.utils.reportError().
Another way is to log to the (native) console from which Thunderbird is started. This is done through dump().
The newest way to log messages is using the Log.jsm module. It is a very nice wrapper around the different logging methods, and my preferred way to log messages in Thunderbird.
Thunderbird Developer Tools
As you haven't linked to the reference, I'm not 100% sure, but I think you mean the possibility to remotely debug Thunderbird through Firefox. You don't need to download anything to use this, it is already integrated in Thunderbird.
execute Javascript that can reference my extension
Debugging Thunderbird remotely through Firefox gives also access to the console and Scratchpad in the Developer Tools. Both should have also access to the add-on.
You may also want to look at the Tiny JavaScript Debugger. It also allows the execution of arbitrary code while debugging.
