The alternator will not cause a no crank condition unless the battery is down because of faulty charging. If the brake light switch is bad, draining the battery, can cause the no crank and no stop lights.
Why did you replace the alternator? The engine spins over? The brake lights are not on a battery bus, should only work with the key on. The rear SAM module may be keepng the brake lights on, and it plays role with the fuel pump.