Is it possible to replace a phone battery with a fully charged one when the phone is out of charge?

if you phone has a removable battery then yes.... but most phones are going to non-removable batteries

If your phone battery comes out of your phone then yes. Mine does not. I recently bought a battery case for my samsung galaxy s6 and I love it. Gives you an extra charge and can be charged at night and since it's connected to your phone you only charge both at the same time. Does add weight to the phone. Easier to grip and seems safer with the add on case. 32.99. used it on a 6hr bus trip.

Yes.

Shut the phone off before changing the battery.

Yes, but it would be BETTER to turn the phone OFF, and only tuern it on for 5 minutes every hour to see if you haev any messages.

yes