Could Naruto Possess the Rinnegan?

Naruto could obtain the Rinnegan by means of transplanting. The only option would be taking the eyes from uchiha madara whom already awaken the Rinnegan. On the other hand he could not awaken a Rinnegan by just mixing uchiha cell to his body.

Would Naruto get Rinnegan if he got Sharingan?

From observing Sasuke and Madara, Naruto not getting the Rinnegan could be because he lacks the Sharingan. ... Sasuke received Ashura's chakra from Hashirama and the other from Hagoromo himself. And even if it was from Hagoromo by himself, it's more than enough.

Can Naruto awaken the Sharingan?

No, Naruto could not unlock neither the Sharingan nor the Byakugan. It doesn't matter whether he was a reincarnation of Ashura. As a matter of fact, Ashura himself didn't possess the Sharingan or Byakugan, despite the fact that his predecessors had different dojutsu.