Secret behind Obito's Kamui?
I've got a lil theory or take on what's happening, so hear me out before you slaughter me? Obito saw in his sharingan an image which will later transpire as Kakashi stabbing Rin. I don't believe they have a shared link or that Obito being outside the cave played any part in him seeing through Kakashi's sharingan.
I believe Obito awakened his sharingan to protect his friends or more importantly Rin, and that when she was being killed by the owner of his eye, that filtered into his own sharingan, hence why he saw it.
Furthermore, we know the benefits of having Hashirama's dna in your body. But imagine being inside and MERGED with a Hashirama infused creation and you awaken your MS for the FIRST TIME. That surely must do something unique and never seen before in the Uchiha. What if by awakening the Ms within Hashi's unique and healing 'body' it immediately healed and repaired itself?
They say MS begins to deteriorate your eyesight the moment its awakened, so this could possibly be the reason why Obito uses Kamui with reckless abandonment. He is not subjected to the same MS curse as the rest of his clan.
This theory is purely speculation so of course I'm aware there are plenty of holes in it. Just wanted to get some thoughts on this.