Alot of the noise you can do away with by eliminating the actual audio from the footage and recreating it for effect after the fact. This is actually the goal of audio gathering while shooting a movie. Clean dialog and nothing else. You add in everything else afterward.
Since you have almost no dialog, you can eliminate all the background audio and record footsteps in a quieter environment. Play the movie on a tv and record yourself acting out the walking on cement in an unfinished basement wearing her shoes (or have her try to match her own footsteps if your feet are too big
). Add appropriate echo to it and you're good to go.
Same with traffic...record it separately to put the sound in the background so you can control the volume and where the sound hits. This way, car passing sounds can be laid over cuts making them feel more seemless as there's no jarring audio cut to go with the visual one.
As an excercise, take a copy of the movie as you've shot it and eliminate all the audio except the dialog, then layer the sound back in...pick traffic sounds from a long take with lots of annoying traffic noise and lay it across the whole chase. You can now adjust the traffic noise lower to make it seem more background and less obvious. To accentuate a cut, like when she turns around to look, no guy, starts walking, cut to guy behind her...the last cut you could put a louder car passing sound to punch the cut and make it more jarring and scary. When she enters the tunnel, add more echo to her footsteps (the ones you've re-recorded).
For effective use of soundscaping, watch M. Night. Shaymalan's "The Village". In the scene where the camera is just tracking across the forest from outside the fringes of it...with the sound off it's just a pretty scene. All the sound was added to make it sound scary, the creaking trees, the wind rustling the leaves, even the animal noises.
I like to watch movies with my eyes closed every once in a while to see what they did to the audio to make it effective. Especially horror movies. Watch through SAW once with your eyes closed, it allows you to disconnect the visuals and see all the psychology they are playing with in your head. What noises they've chosen to let you hear.
I don't mind the shakiness in the shots (and this is all just personal opinion) but the shots where the camera is beside them walking is tilted to the side. It's hard to carry a camera and walk sideways without tilting it, hence the suggestion about the tripod. The weight of the legs will force verticalilty out of the camera. You can also just rotate the footage slightly and zoom to cover the corners in the editing suite.
I've shot in the snow, I know exactly what you're talking about with the cold...I find that to be a perfectly acceptable excuse.
