I have a couple questions regarding Challenge Claims. I read in 4.3 that, "If an effect has the word "cannot" in its description, then it is an absolute: That effect may not be overridden by other effects".

Is a Military/Intrigue/Power claim considered an "effect" of winning a challenge? For example, if a player had the "Stoic Resolve" plot card,(kneeling characters cannot be killed.) and a player defended a military challenge with their only character, would they have to choose that character to be killed for the claim, or does the plot card consider claims an effect? Same question with winning an intrigue challenge against a player with "The Laughing Storm"(Cards cannot be discarded from your hard while The Laughing Storm is standing.)

You would essentially get to ignore the claim effect. Keep in mind that in your first example, if you had one standing character and one kneeling character, you would HAVE to choose the standing one to die.

Ah, thank you for clearing that up for me. I was just a little confused for whatever reason after reading the FAQ.