He had the Harashima cell graft before he awakened the Rinnegan.edo has nothing to do with awakening the rinnegan tho and he clamied he awakened it before his death
He fought the battle of the end to gain a portion of Harashima's power.
Sometime after he gained Harashima's power, he awakened the Rinnegan.
It's not 100% that you need Senju DNA to awaken the Rinnegan, but Senju DNA is heavily implied as being part of it.
It makes sense since the rinnegan was the Sage's eyes (at least according to Jiraiya) that you might need to be as much like the sage as possible in order to have the Rinnegan.
As for how Sasuke might get Senju DNA, There's the fact that Tobi had access to Sasuke's unconscious body, and he's currently fighting Kabuto who may actually want to inject orochimaru cells into Sasuke, and Orochimaru experimented quite a bit with Harashima's DNA.
I'm ambivalent as to whether Sasuke gets the Rinnegan or not, but i'd be surprised if he didn't.