A/N: This will probably be the Author's Note on Chapter 13, but I figured people might appreciate having it available in the meantime.
Chapter 12 (3): Combat Breakdown
Obito was paralyzed because Hanzo did the 'turn myself into an alternating current' thing he considered at the start of Chapter 11, so when Obito grabbed him it was like grabbing an uninsulated power wire (adjusted up to account for his physiology). Hanzo shot him while he couldn't move, and made sure to hold a skin to skin grip on his hand when he turned around and shot him again.
The large flare of chakra Hiashi sees while Obito can't move was Izanagi being primed despite the physical paralysis.
[The Sacrifice] is the first instance of Hanzo controlling (and sustaining) Yemo / the Anami outside his body. He sent them out in Chapter 11 (see 'plasmodial slime configuration') as pre-emptive counter-ambush. His original plan was to make them shove their way into Zetsu and/or Obito through their feet, in a bid to fuck them up. He expected to lose direct control of them the moment his feet left the ground (as they only had connection with him through his soles at that point, aura notwithstanding). But his long shot with Kurama paid off, so Ninshu gave him the ability to communicate, understand and direct the plasma hive mind seamlessly in real time (the same as he would if they were still in his body where they belong). This led to the effectiveness and utility of the technique scaling much higher than he'd practiced.
Yemo and the rest assembled into the four-armed warrior form kinda sorta on their own, based on the psychic cross-flow with God Dad.
The Adamantine Chakra Chains are something Hanzo unlocked off-screen when he absorbed some of Naruto's blood (it was about a week or two after the Gate of Death thing in chapter 8). He didn't know how to use them until the Ninshu with Naruto and Kushina's hidden ghost in the seal. It turns out they're very intuitive once you figure it out, which will come up later.
Black Zetsu tried to remove Obito from the battlefield, because he doesn't have the Byakugan so he didn't know for sure if Izanagi had been cast.
Gama is Gama. Naruto rode it because it's big naturally, almost as tall as Hanzo, and toads/frogs have very big tongues relative to their bodies. Also, it's got the same plasma souls as Hanzo himself, because that's how he made them to begin with. Ninshu allowed Hanzo to control him through Yemo (as sort of a relay).
Trito's Tribulation is literally just the anami treatment, except he's using his own plasma cells and supercomputer brain to make it happen.
Yemo's Sacrifice is basically disintegration via telefragging by turning collisionless plasma into regular plasma while inside the other guy.
The forcefield was by locking the energy state of the air inside the plasma field.
Hanzo et al. didn't 'let Obito go', they settled for driving him away because they couldn't contain him. Even so, Hanzo tried to trap him with the chains because, as others have noted, Obito needs to turn solid to use Kamui. Alas, the guy is a flighty bastard so he ate the telefragging damage with his plant/Hashirama side that can recover from having his heart destroyed (despite that NOT being his Hashirama/plant side) and ran away anyway. (Note that he could have run away intangibly and kamui'd after he was clear of Hanzo's trap, since he can still flicker faster than Hanzo can send the chains after him, but he decided to grandstand instead).
How much longer Hanzo et al. could have kept going depends on how much biomass Hanzo has left after needing to recover from the damage of the Eight Celestial Gates (and being burned to death), and Hiashi's ability to endure further abuse of time dilation magic. Exploring the former, at least, is still pending.