Suppose I have Daring Thief out. My opponent attempts to kill it via Doom Blade or another kill spell. In response I untap Daring Thief with Thassa's Ire, and attempt to gain control of, say, my opponent's 4/4 token. There are no responses, and the exchange happens. I now control a 4/4 token, and my opponent controls Daring Thief.
Does Doom Blade "follow" Daring Thief and kill it on my opponent's side of the board?
Yes, if you exchange control of Daring Thief with another creature while a spell targeting Daring Thief is on the stack, the spell will still be targeting the Daring Thief when the exchange completes. The Thief is still the same object when the exchange happens, it just has a different controller.
Yes.
You might be thinking of the following rule:
400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence. There are seven exceptions to this rule:
There is only one battlefield, not one per player.
400.1. A zone is a place where objects can be during a game. There are normally seven zones: library, hand, battlefield, graveyard, stack, exile, and command. Some older cards also use the ante zone. Each player has his or her own library, hand, and graveyard. The other zones are shared by all players.
As such, Daring Thief never changed zone, so 400.7 does not apply, so Doom Blade never lost sight of it. Doom Blade doesn't care who controls its target, so it gets destroyed.
