New to playstation and have a few questions (i couldn't find them in the FAQs)

So this is my first ever PS console and i need help with a few things:

1) Does the PS4 controller automatically stop charging when it's fully charged via wall charger? If no, are there any LED indicators to let me know the controller is fully charged?

2) If i delete a game on my profile will other profiles be effected by it? e.g. if i delete Persona 5 from my profile will it be deleted from every other profile.