OK, so I know from the FAQ that here is the Framework for the end of challenges:

1. Determine Winner of challenge.

2. Challenge result is implemented. (Does this include the Claim?)

3. Reward for unopposed challenge is awarded.

4. Renown is awarded

A) There doesn't appear to be a window for Player actions between here and passing to the next player becoming the Active player, or ending the Challenge phase? Is that correct?

So here's the main questions:

B) When do/can I activate Areo Hotah ( Response: After you lose a challenge by 4 or more total STR, kneel Areo Hotah to choose and kill 1 character. )? Is there an opportunity to use Areo and then kill him for Military Claim? This came up specifically in a situation including Fury of the Stag ( After you win a Power challenge against an opponent with a Targaryen or a Martell House card, choose a character controlled by that opponent. Take control of that character until the end of the phase. ) where my opponent was going to take Areo with the Plot effect, but I wanted to use his ability first.

C) What is the timing for an opposed Military challenge, in which I win, and my participating character has Deadly? Meaning, does my opponent kill an opposing character because of Deadly AND then fulfill claim, or can they be the same? And more importantly, what is the timing of the kills?

Next, the question is regarding Ice Fisherman ( If it is Winter, Ice Fisherman gains a Power icon and: "Marshalling: Kneel Ice Fisherman to take 1G token from any opponent's gold pool, and add it to your gold pool." ) So if I control Ice Fisherman, and my opponent is Active Player:

D) I can or cannot steal gold from my opponent during his marshalling efforts? I assume, if he is the active player, then he gets the first chance to take an action (likely choosing to spend all of his gold before I get a chance) and marshal a character/location/attachment, but then I get a chance to steal remaining gold, correct?

Finally: The question is about text boxes in general.

E) Are the text boxes of characters in the discard or dead pile active? For example, does an Intolerant Priest in the dead pile gain a Holy crest if I control 3 Asshai characters in play?

2. Challenge result is implemented. (Does this include the Claim?)

Yes

A) There doesn't appear to be a window for Player actions between here and passing to the next player becoming the Active player, or ending the Challenge phase? Is that correct?

Looks that way. There is a player action window before the activ eplayer declares their first challenge though

B) When do/can I activate Areo Hotah ( Response: After you lose a challenge by 4 or more total STR, kneel Areo Hotah to choose and kill 1 character. )? Is there an opportunity to use Areo and then kill him for Military Claim? This came up specifically in a situation including Fury of the Stag ( After you win a Power challenge against an opponent with a Targaryen or a Martell House card, choose a character controlled by that opponent. Take control of that character until the end of the phase. ) where my opponent was going to take Areo with the Plot effect, but I wanted to use his ability first.

Responses don't need player action windows to play so Areo Hotah can respond when the trigger allows him to. The order of responses is as follows:

1/ Saves/Cancels to that trigger

2/ Passive effects.

3/ Other responses

In this case the trigger is winning/losing a challenge. If nothing cancels the winner or loser then next is the passive effects - in this case fury of the stag. After this is responses such as Areo's. So in your example he is "stolen" by Fury of the Stag before he can use his response.

C) What is the timing for an opposed Military challenge, in which I win, and my participating character has Deadly? Meaning, does my opponent kill an opposing character because of Deadly AND then fulfill claim, or can they be the same? And more importantly, what is the timing of the kills?

Deadly is a passive effects, so it is in step 2 as above. This is after challenge results afre implemented (including killing a character for claim). This means that if you only have one defender against a miltary attack with more deadly characters than you then you should choose and kill your only defender as a result of claim. That way there are no defenders left to kill as a result of Deadly.

Next, the question is regarding Ice Fisherman ( If it is Winter, Ice Fisherman gains a Power icon and: "Marshalling: Kneel Ice Fisherman to take 1G token from any opponent's gold pool, and add it to your gold pool." ) So if I control Ice Fisherman, and my opponent is Active Player:

D) I can or cannot steal gold from my opponent during his marshalling efforts? I assume, if he is the active player, then he gets the first chance to take an action (likely choosing to spend all of his gold before I get a chance) and marshal a character/location/attachment, but then I get a chance to steal remaining gold, correct?

Yes that seems correct. If he can play an action before you then he could spend some gold before you steal it.

E) Are the text boxes of characters in the discard or dead pile active? For example, does an Intolerant Priest in the dead pile gain a Holy crest if I control 3 Asshai characters in play?

No, unless they (or a triggered action) specifically allows it (such as Lady Stoneheart). Or else cards that give +1 STR to characters in play (such as Hoster Tully) would do so from the dead pile. Which would be insane!

A) Correct

B) Note that the entire block of framework events happens without an opportunity for responses (Except save/cancel). Once you've awarded renown, you then move on to step 4: Passive abilities (now triggered) are resolved. This is where Fury of the Stag would come into play (or, for instance, deadly). If there are two conflicting effects, the order in which they are resolved is determined by the first player.

C) Kill a character for claim (Challenge result is implemented), award renown, then deadly (and any other passives) trigger.

D) If you are the first player, you get the first opportunity to act, EVEN if he is the active player. You can steal a gold before he has a chance to spend any. If you are not the first player, he gets an opportunity to take an action (say, spend his gold) before you can trigger the fisherman. Note that kneeling a cost-reducer counts as an action... so if he's forced not to use them for fear you'll steal gold, you've still accomplished something with the fisherman (and haven't knelt him!)

E) Unless explicitly stated, cards do not interact with cards in the dead or discard pile, and cards in the dead/discard pile don't interact with cards in play (or each other).

Thank you both, you rock!

D) -istaril and Marshall Lambert appear to give different answers here. I agree with -istaril's reply, with the addition that the action window proceeds clockwise from the First Player, so you may be able to steal gold if you're not the first player, just as long as you're not sat to the immediate right of the first player.

@ above

That's right. My answer was badly worded. I was agreeing with the statement that you can steal gold during the marshalling phase even when not your turn. First player gets their turn first and therefore gets first marshalling action. Even when not their "turn" the first player gets first matshalling action in another plaers turn.