Are the gods immortal?
Of course. That's what makes them gods. Not immortal, not a god.
By default, I'd assume so, and I'd have some difficulty if a author made casual mention that they weren't.
By default, I'd assume so, but it wouldn't be a big deal if an author said they weren't; I'd just go on with the story.
I assume not but am willing to hear arguments to the contrary
I have no default assumptions in this regard.
What ought to happen to a dead god?
Same thing as to dead humans in that cosmology
An afterlife of some sort, mixed in with humans
An afterlife of some sort, no humans allowed
Roaming the world(s) as a zombie god
Something else I will detail in the comments, especially if it is amusing