The only reason that I recommended turning it off (either in setting or by airplane mode) is to conserve battery life.When my phone is in dual sim mode the battery discharges faster and needs to be recharged more frequently.
On the iPhone after you add an Esim there will be a section under cellular labeled cellular plans which allow you to turn on or off the cellular plans. If you only have one cellular plan on the phone then airplane mode will turn it on and off.
I would suggest turning off Public Mobile in your phone settings while you are in europe. If you leave it on your phone will be constantly searching for a Public Mobile network to connect too wearing your battery down faster.
To change your phone number you login to your selfserve account. On the overview page in the My Profile box you can click on 'Change Number' or you can go to 'Plan and addon" tab and click on 'change phone number' there
If you go to settings > General > About scroll down it should say ‘No SIM restrictions’ next to ‘Carrier Lock’. That means your phone is already unlocked It is normal for the phone to show the current/last carrier it was connected to next to 'Carrier...
