Do JSF / PrimeFaces support well HTML5 Storage methods (sessionStorage / localStorage)?
Not out the box. But you can just wrap the necessary HTML/JS code in custom JSF components. One of our previous interns have done that. You can find it in this Google Code repository. Check the POC-SessionStorage part for the source code. You can find a writeup in this thesis.
To the point, it's merely a matter of generating the right HTML/JS code and hooking on JSF ajax events.