I don't need a lecture about childhood development and you are not on the receiving end of a child having nightmares about being shot dead and run over. If you were perhaps you would have a little more empathy for the situation we are in.
now, who's being patronising?
For goodness sake, I was simply pointing out that a lot of this behaviour is something that happens when kids go to nursery and school. It's not a matter of being harsh, but simply reality. You think I haven't had to deal with any of those things? I have a 4yo, so I'm fairly well versed in some of the things he's heard and seen from other children in reception this year (as well as nursery last year).
Your child is going to hear loads of things from other children over the school year. Some good, some bad. Some namecalling, some things you'd rather they not hear at all. The staff at nursery can only do so much. Talk to your child at home about anything that is upsetting them and simply make the nursery aware of the problems in a calm manner, without demanding any specific information regarding said child.
Not any problems with my moral compass thanks. 
You've already said the nursery have told you they have spoken to the parents, so obviously they are dealing with this. You've mentioned behaviour charts and special toys as rewards, so they seem to be working on behaviours. Based on your comments, they're already aware of the namecalling and biting. And I find it hard to believe they wouldn't have seen the shooting games, but you can always mention it. It's possible there are SNs in play here as well.
The Santa thing is irrelevant. Sorry, but it is. At some point, some child will say it in front of your child, mainly because they'll have heard it from someone else. That's life. You just discuss it with them however you are most comfortable, either telling them the truth or reinforcing the Santa myth. Not worth mentioning to the teacher IMO. I think you might look a bit hysterical bringing that up to the teacher.
Please let's stick to that.
Check the top of the page. AIBU. Not everyone is going to say exactly what you want.