Well, if the battery status says 100%, I would take the battery out. The computer will still work because it is plugged into the charger which is also a power adapter. So, its not that good to leave the battery in it all the time while it is plugged into the wall with the charger because it will shorten the life of the battery.
Not unless you want to greatly shorten the life of your battery.
I would not, it should have a charge method for. If so then make sure you charge it up then take it off of the charger.
I am a ex prat and bad thing, but new battery
It is actually better to leave it plugged in so the battery does not go dead.