Um, let's start at madara. He is/was Indra's reincarnation, so he has indras chakra. And how did he get ashura's chakra? By implanting senju cells. That combined gave him the rinnegan.
Wrong, but you were close. You were on target with your point about Madara. Madara was Indras reincarnation, thus he already had Indras CHAKRA. The sharingan had nothing to do with that. Madara could have been born blind, but would still have been Indras reincarnation, thus had the chakra.
What you are mistaken about is the senju part. Madara took HASHIRAMA'S dna. Hashirama was Ashura's reincarnation. If Madara had taken the dna of ANY OTHER Senju, he would not have unlocked the Rinnegan. Only when the chakra of Indra and Ashura SPECIFICALLY mix, will the user unlock the Rinnegan. In other words, only a very VERY select, destined, few will have the OPPORTUNITY to unlock it. Right now the only people we know of who have the possibility/opportunity too unlock it are Madara, Hashirama, Naruto and Sasuke since they are the reincarnations of Indra/Ashura. Everyone else is shit out of luck.