Hyuuga and Uchiha clan family roots?
The Uchiha are not a descendant clan of the Hyuuga. That was speculated by Kakashi in the very early stages of the series.

Much more information was revealed afterwards (SPOILER! Hover to view):


   Both the Uchiha and the Senju (The clan of the First Hokage) are descendants of the Sage of Six Paths, who created ninjutsu and had the powers of a god.


Also


   A Rinnegan is the next stage of evolution after the Mangekyo Sharingan, so you can't really use them both at once on the same eye. The Byakugan has nothing to do with it and is a completely separate technique.