You can use Bran to give a challenge icon to a standing Direwolf. The latter part of the ability is not dependent on the resolution of the former part, as there is no word "Then" in the ability's text. (This is similar to Castle Black.)
You can pay 1 gold to gmake one participating character gain renown.
Which lasting effect applies to which character is determined upon resolution of the event's effects. Subsequently changing the STR of affected characters will not change the lasting effect that applies to them. Lasting effects are static, not dynamic.
If you use Flea Bottom to put Arya Stark into play, you can discard her duplicate to prevent her from being placed on the bottom of your deck at the end of the phase.
If you use Flea Bottom to put a unique character into play as a duplicate, that duplicate will be placed on the bottom of your deck at the end of the phase. If it leaves play before that, nothing will happen. The duplicated character will not be affected.
Hello ! What would be the interraction with Waking the Dragon ?
Both delayed effects attempt to resolve "at the end of the phase", the first player chooses the order in which they resolve.
Interestingly, this has the same "Card design by..." text as the first edition version, but the effects have nothing to do with each other.
I've heard that this was supposed to be the original design for the 1.0 card, but it was deemed unbalanced, and they reworked it into something completely different. And this 2.0 version is actually true to the original concept.