Does the cost of living get less expensive when you don't own a car?
No, you've got it backwards, a car-free existence is more expensive. I live in Manhattan, so I can tell you how expensive it is to live here. It costs a lot less money to live in some non-walkable place in the midwest and own a car or two.
I also lived in Arlington VA, and it was walkable but much more expensive than most other places in the USA as well, and you still needed a car to drive to work.
For whatever the reason, packing in people so close together that car-free life is feasible also has the effect of raising the price of everything else, and thus we can only conclude that densely populated areas are economically inefficient.