Is it possible for a Senju to awaken a Rinnegan? Suppose that a Senju got an EMS from an Uchiha and implanted it, would he also awaken the Rinnegan just like an Uchiha did, given that the ratio of Uchiha and Senju DNA is different than if an Uchiha awakens it?
Answer given by @Ero Sɘnnin in the comments.
Not every Uchiha is able to awaken the Rinnegan. Nor can any common Senju.
It is not the Uchiha:Senju DNA ratio that governs the Rinnegan's existence. It is the Asura and Indra chakra mixing.
Also, I think the chakra ratio doesn't matter as we see in Madara's case. Although more chakra can maybe speedup the process of Rinnegan formation.
It is never stated that random Uchiha or Senju awaken the Rinnegan via the reincarnation process, but rather, the Indra and Ashura reincarnates.
I highly doubt the portion of chakra you have between the two matters, as Madara is 100% of an Uchiha and merely biting a Senju still got him a Rinnegan, but awakened it when he was very old. It is not the amount of chakra, but rather, the fact that you possess Indra and Ashuras chakra.
It is safe to assume, as Alfro said, if you have more chakra of a Senju, the process will speed up.