Many video games including first person shooters help kids strenghthen hand eye coordination skills. But the problem is kids aren't really supposed to play them in the first place , most fps games are rated mature and there's a reason for that so if you don't want your child exposed to violence then don't give them access to the game.
While it is true that many major titles like Call of Duty or Battlefield are arcade-like and not representative of reality, other games like Arma or Red Orchestra 2 are more accurate. It's not like the glorification of war is exclusive to video games; it has been in every form of entertainment media to date. I can't even begin to count the number of movies that fit the bill. Some games like Spec Ops: The Line actually provide really insightful commentary on war. Others are outright shy down by critics from the community for being too accurate.
It makes the kids want to be like the people in the game. Have you ever heard the stories about those kids shooting people? The rate of kids killing people are rising and i think it is because of games like these. I would let my kid play these games until he was probably 15
No they are not healthy to kids for a number of reasons (not to say that there are no positive aspects to these games), firstly they give kids a false sense of what real war is like and therefore desensitize them to that kind of thing, secondly its gives them false ideas about the physical limits of the human body (in no way am i saying this applies to all kids, but mostly younger ones), for example healing quickly after being shot, being able to be shot with multiple bullets and still survive and lastly i must say that this definitely does not apply to all kids, but it does to the majority of them, so no, i do not think that they are exactly healthy for kids.
First person shooters aren't exactly "bad" for kids, they provide entertainment and there's no evidence to support that they negatively affect mentally stable children. However, they're not "healthy". A child who plays fps's isn't going to be any healthier, mentally, emotionally, or physically, than a child who doesn't. Playing shooters isn't hurting children, but it's not improving their health.