Regarding your Sukka, the walls do not necessarily have to reach the Schach. However, in this case, the wall should be directly underneath the Schach with no horizontal gap between them at all.
Assuming he only pours the water into his mug only when it is cold and clean, the kettle is not Treif. The steam rising off of his mug to the kettle will not forbid it. However, this arrangement is less than ideal.