Is it possible to hack camera calibration without having access to the camera?
If I understand correctly, you don't need the intrinsics or extrinsics to achieve that, if a top-down view is all you want. You could basically define 4 points on your parallel lines and then warp the entire image into a canonical view (say $\{\{0,0\}, \{480,960\}\}$). 

To do that in OpenCV, all you need to do is compute the homography using findHomography as described here. Then, use warpPerspective (look here) to warp your images.

The only thing you need to care about is the selection of the dimensions of the fronto-parallel view that you are after. It should more or less match the aspect ratio of the original image region.



Regarding the second part of your question: Yes. First of all, let me remind you that many works such as this choose to ignore the distortion estimation. But, I can think of two ways to accomplish that:


Use the road lanes itself to correct the distortion: In the fronto-parallel view, create another perspective warp so that the lanes themselves would be vertical. In this setting, think about the lane lines as the vanishing lines in the image (as in a perspective view, they always do). From that, you can even calibrate your camera. Look here, here and here for example. 
The cars should be following more or less a straight/smooth trajectory. If you compute the motion of all the cars, they should give you similar motions, parallel to the lanes. That information is again useful in constraining and generating a transformation, which would warp your image to a reduced distortion one.