Battery in the notebook?

When the battery is full but is still connected to the mains. Is it then supplied from the mains or from the battery?

If the battery is in, then the notebook is powered by the battery. The electricity supplies the battery.

Should you disconnect the battery from the power supply when it is fully charged?

Yes.

Leaving the battery permanently on the power will destroy it faster. You can remove the battery and store it elsewhere and if there's a power failure you are just unlucky when you are at the computer.

Therefore, it is the better option to let the notebook run on the battery in a controlled manner.