Well, though you do get the Rinnegan if you mix first and second son DNA, it apparently doesn't happen right away. We know that Nagato got his fully when he was under deep emotional stress (kind of like how the Sharingan evolution), but we don't know the specifics of how Madara got his, just the he got his later in life.
Obviously, with Danzo as my reference, if you are not a direct descendant of the Sage, you cannot hope to get those eyes, unless there is theft involved. Even if you do combine the first and second son's DNA, it won't work, bud. So that ultimately means that Danzo always was a reject, as one can see with the scar on his face (chin).
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There might even be different reactions to mixing, say, a Hyuga's blood and a Senju or a Uzumaki and a Uchiha; we don't know. We won't know how much Kishimoto has evolved these mythologies until he actually does it.
On other note, there is a Byakugan theory out there that's really well-done, though it does sort of inch towards the crack border. Here it is:
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