No. He would see himself as a hypocrite if he didn't.

fun fact, before the Batman tv series from the 60s, Toho expressed interest in a Godzilla vs Batman. Batman would have been battling a mad scientist with a weather machine while Godzilla would battle a monster under the scientist control.....however the batman stuff was scrapped and but the weather machine subplot was kept for the movie Son of Godzilla.

Probably, because then he'd be rid of the Joker without being responsible for it.

No he saves lives even the bad guys

Batman begins He is not going to kill him or save him

He would probably think there are civilians in the building, and stop Godzilla from blasting the building and in turn not harming the Joker, but if there wasn't a building there, and the Joker just happened to be standing there, I dont think Batman would do anything. Batman, I dont think, would want any part of the Jokers death, and heck if his sidekick getting killed isnt enough to make him kill him I dont know what would make him feel compelled to do so