I see what you mean. It doesn't look like anything has happened in the quoted passage that would suggest he was dead. He is certainly disorientated. You have two choices.
Choice One
"I had not seen him approach, and the sudden appearance made me start." This happens to people all the time when they are concentrating and don't hear or see the things going on around them. But in this particular case, he was only putting a key in a door. That doesn't take much concentration, particularly if you have done this 1000 times before. If you allow me to speculate just a little bit, it sounds like an old fashioned boarding house. It would be hard to sneak up on someone.
Choice Two
Once inside the room, he runs straight to the back room and checks the door, but does not know if it is locked. Then comes back and asks for a favor. It's really odd. It's a hint.
I would say c since he asked in the end sentence. <span> "And so I leave it with all of you: Which came out of the opened door - the lady, or the tiger?"</span>
