Continuing the work of a deceased author... I'd say yes. It's not like it hasn't been done before. Does the word "collaboration" ring a bell? Sure, both authors could be alive, but it's not any different in my humble opinion. I wouldn't stop anyone from continuing the work of a great author. It's just like letting someone continue sculpting a statue a great sculptor had started but now is unable to finish because of death. You don't know the exact details the sculptor had in mind, but you know that the human body has two hands, two feet, a nose. You just have to rely on the details the sculptor had left in order to finish the statue closer to what he had in mind.
As for ethics, I'm pretty sure every author wants their work finished. If at his death bed he says he doesn't want anyone to finish his story, then the story shouldn't be finished. Though I'm most certainly sure art is different from science (for science has laws, and art/literature is an expression of one's soul where anything goes), I agree that whoever continues the story shouldn't get any profit from it. But if he deserves it, why not? If he did justice to the story, why not tell him so instead of telling him what he should and shouldn't do? Who knows if someone had snuck a paragraph into one of Mark Twain's stories without him noticing it? Would you feel differently about the story knowing someone added that paragraph? That's just silly.
I believe it would be an honor if someone asks me to finish his dad's/partner's/grandfather's/mother's/grandmother's/sister's/brother's work. That's not to say I will do it. If I'm comfortable enough about the story, I will. If I know I can, I will.
If I will continue the work of a deceased author, I'd re-write it in my own style. For if I just continue where he left off, there will be a very distinct difference. There's just no way anybody could write in someone else's style. What if the sculptor uses his toes in sculpting? If I'll re-write it, he same things happen, although it would be written in my own words. Like I said, collaboration.
Rad