Minato was using KCM before that moment when they bumped fists and Kyubi wasn't in a cage like where he is supposed to be like in Naruto's caseThis question was already answered by the manga. Kurama is two halfs. They both believe in Naruto so when they bumped fists they were already ready to help him.
You could possibly be right, but despite that what you said mean nothing because we will probably never know what was taking place in the Reapers belly. All we can do is guess and speculate, you can't just say Minato used the Torri seal as if it's fact .Actually I believe that Minato tamed the Kyuubi inside of the Death God because we know once sealed inside your souls are in eternal battle.
As for Hashirama not knowing Minato we have this explanation :
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Hashirama: "I thought you and Saru were sealed together"
Mianto: "Nope the chain of events unfolded a bit differently"
Meaning that Yondaime and Kurama were in a different dimension and Sandiane and others in another one.
Also about Eternal battle - a member in past argued me that if they were figthing a never ending battle then how did Minato win?? Well answer is simple
Minato used theYou must be registered for see linksand Kept Kurama under wraps. They weren't pals [like Naruto and Light-Kurama] sinceYou must be registered for see links.
It's not like Minato is a TNJ God But Naruto is one he made the kurama inside minato believe in Naruto and help him and I think while Hashirama was telling his story Minato beat the crap out of yin kyuubi since he has Infinite Chakra