A: Yes, the Bat Signal is straight-up traded for Batman, and your opponent could choose to simply exchange the two cards back on their turn. While this may seem slightly pointless, there are a couple of reasons this would be useful.
The optimal use, of course, would be to use the Bat Signal to call Batman over to you in order to win the game, in which case, future trade-backs would be moot. You may also wish to call Batman over to you because he has the ability to trash one of your Villains. He could then go to back to other parts of Gotham on your opponent’s turn, no skin off your back, especially since, should you need him on your next turn to trash another Villain, you could exchange him back again.
Think of the Bat Signal as being almost as though you have Batman on call whenever you need him (on your turn). Sure, this keeps Batman running all over town, but, hey, he’s a popular guy!