many had similar problem in the past putting the blame mostly on the adapter itself.
If it's a sealed box, it can't be a wrong charger.
another possibility is to check with the Lighting cable as well. Some claimed replacing the Lightning cable resolves the problem.
Some did a reset and it works as well.

what is there to penat?
Just hook up to your iTunes and it'll backup itself.

There is nothing bad about letting your battery discharge till 0% anyway.
I've done that with all my iPads and iPhones.
The battery acts as a vessel to collect juices for your device. Similar to barrel of wines/beers.