How did C.C get her scar?
Firstly, C.C. did not kill the Nun. The Nun's intent was to force C.C. to take her Code, and once C.C.'s Geass matured, attacked her to make her do so, and then killed herself. C.C. receives her scar from the Nun, during that altercation. 

Presumably, the Nun tried to draw a Geass symbol, but it was done imperfectly. 

Presumably, the scar persists because it was one of C.C.'s features before her death, so it doesn't go away completely.