In the Name of God, the Compassionate, the Merciful.
Hi!
A: I’m quite sure you didn’t kill him.
B: Really. Are you quite sure?
A: Really. The doctor’s not dead. You didn’t kill the doctor.
The doctor’s not dead
What’s " 's" short for here, has or is?
Thank you
In this case it means ‘is’.
Ask yourself which form of “to be” makes more sense in the sentence.
The doctor is not dead.
or
The doctor has not dead.
Compare to:
The doctor’s completed his examination.
The doctor is completed his examination.
or
The doctor has completed his examination.
In your previous question about the doctor’s office, the apostrophe indicates possessive form of doctor, not a contraction.
Thank you so much, NearlyNapping 
Very nice.
As for the first one I would say: ‘He’s lucky’. As for the second ‘what’s’ is not short for ‘has’, however, I think you meant: ‘The doctor hasn’t died.’ Never mind, it’s just a little typo.