Given that your source says that it was published in The New Yorker, I imagine the author is American.
In Europe, floors are described as follows: ground floor (street level), first floor (first floor above street level), second floor, third floor, etc.
However, in America, they call the street-level floor the first floor. So if the publication is American, then "first floor" will be the ground floor.
As for "down the hall" it means "along the hall" in this context. So it's further along the hall, not necessarily at the end. It could be at the end, but all it says is that it's closer to the end than the current position is.
Here's a diagram for example:
Doors 1-3 could be described as "down the hall". Door 5 is right next to him (he is at the X) so it wouldn't be described as "down the hall". Door 4 probably wouldn't be described as down the hall since it's a bit too close, but it could be.