Hmm officially it doesn't look like Sasuke has Hashirama DNA, Sasuke hasn't used Mokuton and hasn't benefitted from anything of Hashirama Cell. Sasuke gaining Rinnengan was from Hagoromo.You mean:
Itachis eyes
Hashis DNA
Haroromos Rinnegan etc ?
OT: If somehow Sasuke will get Kurama this means that Naruto somehow will get Rinnegan or another doujutsu.
It weird Naruto officially got Hashirama DNA in his arm yet no Mokuton or benefit in anyway, that what I call inconsistant since everyone with Hashirama DNA can use Mokuton.
Don't know how Naruto would gain Rinnengan, but Sasuke Sharingan are not being used so there no problem in Naruto using Sasuke old Sharingan, although most likely it is in Obito Lair.