The shooting games have been going on since well before I was a child. Right now it's video games, but with previous generations it was cowboys, power rangers, Flash Gordon for heaven's sake.
His mother just calmly pointed out to you that he plays 18 games? Okay. I find that hard to believe that a 4yo would have not only the manual dexterity but the reasoning skills and cognitive skills to successfully play an 18 game long enough to take anything away from it tbh.
Children do sometimes call names. It's unkind, and I imagine the nursery staff step in and tell them not to or have some type of consequence for repeated offences. In the meantime, work on your DCs confidence and remind her that sometimes people say things that are unkind and untrue. This is something that will crop up repeatedly throughout school, unfortunately.
I am not at all prepared for her to be called fat or have common childhood beliefs shattered by this child who is a lot more wordly wise but my question is what exactly is a nursery able to do? I am upset that my sensitive child is being exposed to things beyond her years unnecessarily.
First of all, I highly doubt the child thought "okay, I think I'll shatter this child's childhood beliefs today." It's highly likely that someone else (older sibling, adult) said something where the child heard it and the child then repeated it in school to other children. These things happen. The child can hardly be blamed for simply repeating something they heard in that regard.
Namecalling, biting, rowdy or "violent" shooting games do crop up in school as young as nursery age, so she's not exactly being exposed to things "beyond her years." This is not Beirut. It's nursery. Kids misbehave, and they slowly learn social skills. If this child is struggling with his social skills, the staff will most likely already be working with the parents on behaviour management already (although again, they cannot discuss this with you).