TBH it's the ability to control Amaterasu as been hinted when he fought Bee and A, when he fought Bee he used his right side if I remember correct to turn on Amaterasu and his left to turn it off, also the same when he fought A and controlled his Amaterasu.
I'm not sure about which eye but he definitely uses one for Each technique and he doesn't have Tsukuyomi U_U, Danzo was mistaken he even noticed that there's a big difference between Itachi's Tech and Sasuke's, but to show that Danzou is a lier, he never actually experienced Itachi's Tsukuyomi so he couldn't have possibly known what a true Tsukuyomi feels like. That's what I think.
Nope sir, not really, it's stated that you need both MS techniques in your eyes, not necessarily Tsukuyomi and Amaterasu, why hasn't Madara show any of the two, and it's also stated on NarutoWikia ( don't trust it that much ) that it only requiers two Techniques and not necessarily Tsuku or Ama .-.