Determination vs shosuro takuro

Can i play Takuro's reaction after he was destroyed with action with determination?

No. Determination has removed the card from the game.

Regarding this: viewtopic.php?f=59&t=74204

Determination reads: Dark Virtue Reaction: After you announce an action, target one of your Samurai: The action cannot have its effects negated or have its targeting changed. After the action destroys any Personality, remove him from the game

Determination's "remove from game" only happens after the personality is destroyed. Takuro's trigger also happens after he is destroyed. Why does determination have priority? Can you please explain? It seems like determination just adds an extra effect to the action, which would trigger after the destruction; I know the situation is different, but I know that cut them off can be used against doomed intentions before the person playing doomed will have the opportunity to take a second action, and it seems like the situation is similar here.

Doomed Intentions creates an additional battle action: 2. Some effects grant an “additional” action of a given type (usually Battle, Open or Limited). When an action or triggered trait grants an additional action, the timing of the additional action is delayed, such that the action is taken once the action or trait that granted it has ended, and the opportunity to play Reactions to the end of the action or trait has passed.

Cut Them Off reacts "After an action resolves", so the additional action from Doomed Intentions is delayed until after this.

The case of Determination vs Shosuro Takuro is very different. Determination doesn't create an effect that triggers after the action resolves, it creates an effect (technically, an indefinite conditional effect) that triggers after the original action "destroys any Personality". Similarly, Takuro doesn't wait for the end of the resolution of the action, he reacts "after [he] is destroyed". This brings us to Timing, Resolving timing conflicts: If two or more reactions, triggered traits, or rules procedures are triggered at the same point in time, follow this sequence from A through F to resolve their order. D. Apply any delayed effects or indefinite conditional effects that resolved previously and are timed to the trigger (such as, “After the next time this turn he assigns…” or "After each time this turn a Personality assigns...") The active player decides the order in which more than one of these effects apply. If different sources each give multiple delayed or indefinite conditional effects to the same trigger, the active player decides which source's effects apply first, but multiple such effects from the same source all happen before such effects from the next source happen. F. Reactions may now be taken following an action round procedure. Starting with the active player, each player in turn order has the option to take a Reaction to the trigger or pass. When all players pass consecutively, no more Reactions to that occurrence of the trigger may be taken.

Determination's ICE triggers in step D. Takuro would trigger in step F, except he's no longer around.

Why can Takuro not use his but noriko can.

Because Kakita Noriko is not removed from the game, whereas Shosuro Takuro is removed from the game.