I don't get the impression that the river flows from the lake (unless I'm missing it somewhere?), so maybe it's just a creation run by pumps, like a giant fountain? Or it comes from outside and the Gamemakers just control it?
She runs for ages on a downward slope without reaching the river the first day. Maybe she's running diagonally, or zig-zagging to through off her scent.
The Careers are up by the lake. I'm guessing that, before she and Rue set the trap, they're down near the bottom, since it takes them a while.
With the red line indicating Katniss's path. The Careers are actively hunting, so their appearances can occur anywhere, regardless of their camp. The Cave seems to be where the rocks come down steeply to the river. I made it a round, dome forcefield for no good canonical reason, I guess. (Quite by accident, I put the fire close to where the barrier is, which would make sense.) The scale would be way off, considering how long Katniss walked, I guess.