We are conditioning out the case where the host picks the door with a car, so there's only two scenarios of equal probability left. Hence 50-50.
Well yes, if you throw out half of the instances where your original choice was wrong, then the chance your original choice was correct will inevitably go up.
Well yes, if you throw out half of the instances where your original choice was wrong, then the chance your original choice was correct will inevitably go up.