Cheat to unlock all gadgets in Batman: Arkham City

To pull up the map, press Back/Select to bring up the map, then you can use Ⓨ/△ to set and clear objectives. Pressing Ⓨ/△ will set a waypoint/objective, then you can press it again to clear it and return to the story objective. The compass at the top of the screen (not visible in detective/thief modes) will also show your waypoint/objective.

I'm going to go out on a limb here and say, no, he doesn't.

The Joker's such an iconic Batman villan, there's no way to kill him. The Joker's an integral part of Batman's universe. As with every comic book character, he can appear to die and then show up in the next issue/episode/sequel as if nothing has happened.

I would like to note, however, that this game was Mark Hamill's last performance as the Joker.