Obito and zetsu
We have all see that zetsu is a loyal mate to tobi, then my theory is that when kakashi and rin left obito it that cave is has try one last thing before he dies. I think that between the instead kakashi and rin left the cave and his death, he has automatically develop the Mangeuko sharigan and activate his special ability the space time ninjutsu and been teleport in the other dimension.
In that other dimension is where he meet zetsu. Zetsu seeing him near to death help him to recover maybe one of the ability of zetsu, obito having his life saved promise to zetsu that they have a new life in the naruto dimension and team together. But the recovery of obito was not complete that why he use to wear the mask.
Concerning the rinnegan that tobi retrieved from nagato corpse was his eye. Remember that he has offer kakashi his sharigan when zetsu have have gone the medical operation to obito maybe have regenerate a new eye from obito cell. Why is it a rinnegan and not a sharigan, remember that madara edo tensei has develop the rinnegan too he can use both EM sharigan and rinnegan at his will. Maybe that's the way to acquire it And the reason why the other eye is still a sharigan is that it damaged or removed and regenerate
Thanks for your attention folks