My friends and I have been playing for a few months and we've run into a few questions along that way. Thanks in advance for the help on the rules clarifications.

1. What happens when your deck runs out of cards in melee? We remember reading somewhere that you reassemble a deck from the discard pile but I can't find where it states exactly what goes into the new deck. Also, if there is only 1 card left in the deck, what happens during the draw phase when you are able to draw 2 cards?

2. When playing with the card "The Scourge" (ODG), do you have to trigger the response (to lose a card and gain an icon back) immediately after the scourge is activated or can you trigger that response later in the phase?

3. When a person is running the seige of winterfell agenda and they win a undefended power challenge, what happens? We currently think they do not gain the power normally gained for unopposed challenges but do get to move 1 power from the opponent's house to their own.

4. Do effects like "Balerion the Black" which says "kneel all .." count as choosing all affected cards as targets? In other words, can something like starfall advisor which says "Discard 1 gold token from Starfall Advisor to cancel 1 event card or character ability that chooses a House Dayne character as a target" cancel Balerion's ability?

5. For the event card "ill tidings," can you play the card (if you dont have a maester/Learned character" and kneel an opponent's maester/Learned character to discard an attachment?

6. What happens if there is a unique character in play with another copy in the discard pile and an opponent activates "Visenya's Hill" to move the copy in the discard pile to the dead pile?

7. Can you play a event card like "He calls it thinking" to cancel an response even if you don't have a character in play to attach the event card to?

1. You can't find this rule, because it doesn't exist. When there are no cards in your deck, you simply cannot draw anymore. If you need to draw multiple cards and there aren't enough, you draw what's left.
2. The opportunity to respond to an effect has to be used pretty much immediately: once all players pass on an opportunity to use a response, the action window closes and the opportunity is lost.
3. You are correct on this one (the amount of power moved is the claim value of their revealed plot, at the time the claim is implemented - most often 1, but other values are possible).
4. There is no implicit target choosing in AGoT. If the verb "choose" doesn't appear, then no card is chosen. Balerion's response doesn't choose anything (contrast with Field of Fire, which says "choose all non- Dragon participating characters")
5. You can only pay costs with cards you control, so no,
6. Nothing happens. The rule about uniques is enforced only when a unique card tries to enter play or when someone tries to take control of a unique card.
7. Yes, the main effect is to cancel. If part of an effect is after the word "then", that part needs what's before to succeed, but the reverse is not true. Note that you could attach it to an opponent's character, and you would have to if the only Martell characters (who could take the attachment) in play were your opponents'.

Take a look at the FAQ, it contains answers to many questions.