I don't think it does make the car accelerate any faster if you add upgrades after the acceleration bar is already full.
The brakes don't really decrease stopping distance but they do get rid of brake fade (the game tells you this in one of those little info bubbles that pops up once and a while). So you can brake hard in quick succession and your stopping distance won't increase from brake fade.
Car customizing is amazingly half assed at the moment. It would be nice to have it redone at some point. Be able to change top speed and traction, have armor slow you down the thicker it gets, lift trucks and cars instead of just lowering them, be able to customize the cool vehicles like Semis or bulldozers, They have so much potential to make it amazing. Hopefully they will.