Bruce Wayne died in Final Crisis as he sacrificed himself to save the world from Darkseid.
batman doesn't die
No.
No
Stephen Batman died in 1584.
Yes, he does die in batman begins because of the train crash.
Assuming there's no kryptonite Batman, obviously
no
no
Batman's parents died in Crime Alley when a criminal shot them.
Yes he did die sadly.
In Batman R.I.P; Bruce doesn't necessarily die, only looses his memory of ever being Batman. (I'm sure there's more, but that's all I've got.)
If you are talking about Batman Arkham City then yes. If you're talking about Batman Arkham Asylum, then no.