The Barriston Selmy who has the text "Challenges: While Ser Barristan Selmy is participating in a challenge, discard a card form your hand to choose 1 participating character. Until the end of the phase, that character gets -2 STR and is killed if its STR is 0." When does this resolve?

So I issue a challenge with two characters strength 2 and deadly, they defend with Barristan, throw a card away and kill my character. Does my character die and leave the game as part of the player action, or are they moribund until the end of the challenge? i.e. I win the challenge, and have more deadly characters so would kill Barristan and get my claim? If I had the ability to raise their strength to more than 0 could I do that to save them or are they moribund before I can raise their strength? Or does my character die on the spot, and I now lose the challenge. Would the dead character's deadly still apply?

How about in defence? Would the challenge be undefended if the character had been killed in the player action before resolving the challenge, or are they still considered to be contributing their strength and die after the challenge resolves? If I am defending and losing a military challenge can I select the same character to satisfy the claim?

Moribund cards leave play at the end of every individual action window. Selmy's ability is a player action, so the affected card would leave play at the end of that action - well before the challenge resolution framework.

So, in the attacking scenario, where Barristan defends, the targeted character would leave play before deadly could resolve. Barristan is introducing a terminal effect. If they chuck a card to reduce one of your 2 STR characters to zero, it dies. Without a dual-effect that both saves them from death AND raises their strength (like Risen From the Sea), there is nothing you can do to save the character... even by raising its strength above zero (because they're technically already moribund at that point - so raising their strength does not save them.) I'm assuming in your scenario that they're chucking TWO cards so that both characters die (otherwise, one Deadly character would still be in play during the resolution framework).

The same is true when Barristan is attacking, if they successfully chuck enough cards to introduce a terminal effect on a character, then that character would die before the resolution and, yes, the challenge would then be unopposed (assuming they'd chucked enough cards to remove all defenders from the challenge).

OK, so moribund lasts for the action, not the challenge. What about the abilities that remove icons? As Martell if you remove an icon mid-challenge (after a character is declared as an attacker or defender) is it too late to remove them from the challenge? i.e. their strength is already in the challenge, and deadly would be part of the conflict resolution?

I am fairly sure that the locations that can give a character +2 strength and stealth, if they are applied after defenders are declared can only be used to give the +2 strength as the defenders are already in the challenge?

Each of the abilities that remove icons will have an explicitly defined duration. Most last until the end of the phase, so if it's triggered during the challenges phase, it would last throughout the entirety of the challenges phase.

If a character is participating in a challenge, removing the corresponding challenge icon has no effect in the current challenge. Conditions for participating in a challenge are checked at the time the character is declared or added to that challenge.

You are correct that adding stealth after defenders are declared has no effect on that particular challenge.