@elmo41683 covered it well
Overcharge is impossible. Your phone was designed to be plugged into a wall charger continuously.
As to turning off this is very Powercore and phone combination dependent.
Turning power on:
- some powercore will turn on when you plug in a cable, some will only come on when you press the button. That exactly Powercore+ 10050 I do not which it does - anyone own and know?
- so what happens is your phone gets fully charged but even though it is fully charged, if the phone is on, it is still drawing power anyway because it needs power itself constantly and does not want to take from its battery.
- if the power demand from your phone is small enough, for long enough, then the Powercore will turn itself off at a threshold which is Powercore dependent. In general, the bigger Powercore turn themselves off at smaller demand than smaller Powercore.
- the reason a low enough long enough power pull turns the Powercore off is for simple efficiency.
So what combined this means:
- none can tell
- you have to try it yourself or find someone who has exact same Powercore and your exact same phone and (it can vary by ROM version) the same software.