No notification if screen turned off on android

If I leave my android screen on, my phone rings when front door rings and a notification appears. However, if my screen is turned off, my phone rings, but I get no notification. Even if I quickly turn phone on, there is no notification.Secondly, the notification only lasts 2 seconds. If I'm not with my phone, I can't get to it, turn it on and see the notification before it disappears.Third, ringing the doorbell makes my android ring, but motion sensor only vibrates my android. Shouldn't it also ring my phone? Vibration won't do any good if I don't have phone in hand.