Why would his eyes disappear
Cause his original body didn't have 'em when he died, his Edo Tensei body did cause Edo Tensei regenerates the whole body.
I'm guessing the face remained there, despite being a post-death addition, because it wasn't actually part of the Edo Tensei. It was an addition by Kabuto after the fact. And whatever Madara did to his body before dying is also compatible with it? Not so sure on all that.
Why would Kakashi and Minato sit there and not try to stop Black Zetsu at any point...
Wut. When Zetsu went to take Obito's eye, Kakashi and Minato both immediately moved to stop him. It's why Zetsu re-merged with Obito's body and said "hey dudes don't kill me I'm keeping him alive lol."
Anyway
Naruto
So when Madara absorbed Sasuke's amaterasu, I was immediately reminded of when Sasuke used it on Tobi, and Tobi managed to extinguish it somehow. So I figured "oh, he musta had a rinnegan under the other side of that mask." Then I thought for another one second and realized that was stupid, he didn't have the rinnegan yet, and Madara's not using the rinnegan here anyway, he's using some kinda weird rinnegan-built-into-the-body shit. So I was left thinking "wait... how DID Tobi extinguish the Amaterasu back then?"
I'm assuming it was Izanagi, yeah? We've seen Tobi with a whole Sharingan storeroom, with like 80 eyes, unless I'm making that up. They showed him re-affixing his mask after extinguishing the flame, right? So he used his non-visible eye to Izanagi-away the flame, then replaced that eye with another one from storage later?