Charging while power is off

Yes, it’s normal. You can’t see if the phone is charging if the screen isn’t on and when the phone is powered off, the screen isn’t active. How would you expect to see if the phone is charging if the screen isn’t active?

No, that image you are showing means that the battery is very low and it needs to be charged. You do not see a charging indicator if your phone is off and you are charging it. That image only appears when the battery is very, very low and you are trying to power up.