I stabbed my Sister in the butt with a pencil when we were really young and it stayed in there till the day she died. After all those years,all she had was a gray mark where it broke off. So I guess a small bit isn't going to hurt you.
Depends on the extent of the wound. If it's something you can patch up, do that. If it needs stitches, go to the ER
I stabbed myself with graphite in the leg when I was twelve. Since the graphite got stuck in my leg and I couldn't get it out, and I didn't want to tell anyone. So I left it, and there was a mark for just under nine months and now all that's left is a scar that you can't see unless you know it's there. There were no other side-effects, but you should see a doctor if you're concerned.