For your first question, SAs are always read from the perspective of who plays the card, unless specified otherwise. If Player A placed the card, then Player A gets to choose the next target.
For your second question, you are correct that the character does not have to be in battle. As a side note, though, note that the default for targets is in play, so the character cannot come from your hand or set-aside. Also note that "your" requires control of the character, therefore if your character is being controlled by your opponent or is in someone else's territory, you cannot target it.
