TN X Skill_A (R) is "Must use ring R and skill_A, one target number"
TN X Skill_A (Ra Y, Rb Z) is "Use Skill_A and choice of ring, if using Ring Ra, TN Y, Ring TN Z, all other rings, TN X"
It's defined on page 178. It should be defined on 13, but isn't.
You can use any number of kata which aren't mutually exclusive in their activation conditions..
If a kata's 'activation' is an action (like Iaijutsu: Rising Cut) then obviously you can only use one at once, because you're only doing one action at a time.
You could not, for example, use Iaijustu: Rising Cut and Iaijutsu: Horisontal Blade simultaneously because both are actions.
In addition to on 'action', you can use any number of 'opportunity' kata whose prerequisites you meet if you have enough * to fuel them all.
You can therefore use Iaijutu: Rising Cut and Striking As Fire simultaneously if you have sufficient success and opportunity
You will never be able to use Striking as Fire and Striking as Water simultaneously because the former requires When you make a Martial Arts [Melee, Ranged, or Unarmed] (Fire) check whilst the latter requires When you make a Martial Arts [Melee, Ranged, or Unarmed] (Water) check, and you can't be doing both simultaneously.
the "spend opportunity under a different ring at double cost" ONLY applies to the generic opportunities on page 18
Fire and Water are an 'opposed pair' who can't do that anyway; 2 fire can be spent as 1 earth or air, but not water.
You could, however, use Iaijutu: Rising Cut, Striking As Fire and Disappearing World style simultaneously, because none of their prerequisites are mutually exclusive - Disappearing World and Striking As Fire both require Martial Arts [Melee, Ranged, or Unarmed] (Fire) .