You don't usually say 'I'm scarred' when referring to a physical injury, though you can say 'I have a scar'. ' I'm scarred' is often used to mean that an event or experience has had a profound impact on you and has changed you psychologically, with the implication that it is for the worse. The scar is mental, rather than physical. You could also say that sthg. had left a scar on s.o., again implying a long-lasting mental effect.

Yes. Eg, I have a scar on my hand.