Yes! Just start Early and leave it open before you go to sleep. You can safely lock your iPhone or you can start the Night Mode.
Just do the edge swipe on the main screen where all your alarms are displayed.
If you are using the Night Mode, the screen will be on overnight. That will consume a lot of battery power. That is why it is highly recommended to leave your iPhone plugged. If charging is not an option, just start Early and lock your phone. This will preserve battery power.
Yes, but you need to check theese two things:
1. Go to "Settings" than to "Notifications" on your iPhone and make sure that notifications are allowed for Early.
2. Make sure that your iPhone is not in Do Not Disturb or Silent mode.