Few thing that might make this extension better

[Armalon]

It's a good written extension, thank you for this work! Really enjoyed watching your code, I'm not so qualified to write such a nice and readable scripts. Maybe I'll learn something from you.
Besides praising your work I wanted to notice few things.

1. This extension for FF might be easily transformed to Chrome compatible. I'm sure you know that, but this is for the rest of the visitors.
2. The major thing about solving Recaptcha is calling a method that a target site creator marked as a callback. It's important, because in many situations there might be no "Submit" button, or it's disabled until a Recaptcha is solved properly and therefore a callback function called.
   But there is a solution that I noticed in another similar so called "AntiCaptcha plugin". Thay scan a "___grecaptcha_cfg" global object for this callback function an run it after putting a gRecaptchaResponse into $responseDiv. And it works quite well.