When you install an EyeX eye tracker (EyeX Controller or Steelseries Sentry) the bundle with the device driver also installs a software called the EyeX Engine. By default this will then always run on the computer where the eye tracker was installed. The EyeX Engine will automatically take care of screen setup and calibration, and currently these configuration applications are only available in English.
If you are writing an application that is to be used with an EyeX eye tracker, your users will have EyeX Engine running on their computers, and have calibration handled by the EyeX Engine.
So, no matter if you select the EyeX SDK or the Gaze SDK, your users will have the calibration handled by the EyeX Engine.
It is possible to implement your own calibration application using the Gaze SDK. There are some samples in the SDK package how to do this. But even if you run your own calibration application, there are occasions where the EyeX Engine would overwrite the calibration your application did with the latest saved calibration profile in the EyeX Settings. The EyeX Engine loads the calibration profile when the user logs in to Windows, and whenever there is a system event that invalidates the current profile. This does not happen very often, but we do not have a specified list of when this will or will not happen.
For the second question: yes, you can apply for the Tobii Gaze SDK as an individual developer.