Does this mean that she is trying to get a date with him?
Yes, or go home with him.
This is not an English sentence. You could say 'tries to score with the man' if you mean trying to pick him up. You can't score someone unless maybe you're grading them on a test or judging a sports competition like figure skating or diving.
This is definitely a valid sentence in English; perhaps you haven't heard it before though?
... or make deep scratches in his skin.
Please learn some English, DL!
I haven't heard thay phrase in years
Danish men are notoriously non-committal