I may have missed something important with regard to "Any Phase:" abilities, so bear with me. The actual event in question took place during a two-player game, but I'm having difficulty locating the cards necessary to reconstruct it, so I'm just posing a hypothetical that should answer my question.

Let's say I'm in a three-player game, playing Stark, against a Lannister and a Targaryen deck. My Targaryen opponent is the first player, and the marshalling phase has just begun. He also has Viserion on the field, and he is gunning for my Robb Stark (LoW). Lannister's revealed plot is Hear Me Roar, so Robb is currently at -1 STR. Targaryen plays Dragon Attack from his hand. Is there an opportunity at all for me to name "King" as the target trait of Robb's ability, saving him from the Targ event?

In a similar vein, is there any difference in the outcome if it is a character ability vs. an event doing the killing of Robb (e.g. the Targ player decides not to use Dragon attack, Lannister starts his turn, plays Ser Ilyn Payne and then uses his "Marshalling:" ability)?"

Card Text:

Robb Stark
(4) STR (3) Military Power Noble
[stark Character]
Lord. King.

Renown.
Any Phase: Name a Trait. Until the end of the phase, each character you control that has that Trait gets +1 STR (+2 STR instead if it is Winter). Limit once per round.

Hear Me Roar (5) (3) (1)
[Lannister Plot]

House Lannister only.
Characters without an Intrigue icon get -1 STR.

Dragon Attack

[Neutral Event]

Any Phase: Kneel a Dragon character to choose a character. Until the end of the phase, that character gets -2 STR and is killed if its STR is 0.

Ser Ilyn Payne (4) STR (2) Intrigue Power
[Lannister Character]
Knight. Ally.

Marshalling: Kneel Ser Ilyn Payne to choose and kill a character with STR 2 or lower.

The Targaryen player can play the first action because he's the First Player. That means he can trigger a "Marshalling" or "Any Phase effect" before you can (doesn't really matter if it's an event or a character ability). If that effect kills Robb Stark, he will leave the board before you can even trigger his ability. In other words: in this scenario you cannot save Robb Stark unless you use a save effect that also removes him from Dragon Attack's terminal effect.

Specifically, when someone plays a card or triggers an effect, it must be resolved completely before any other card or effect can be played. You cannot interrupt the initiation and resolution of any effect in this game. (NOTE: The exception to that general statement is the use of Response effects that use the word "save" or "cancel." These are the only interrupts in the game.)

So, once your opponent has played a card (or triggered the effect of a card in play), there is nothing you can do about it, except "save" the object or "cancel" the effect.