I want her to be a psychic girl with telekinesis powers but the idea is she is supposed to stay young but I can't make her a little girl because it might not suit well hitting or punching an underaged girl or killing her would be too violent. Even if I made her 17 would still be pushing it. So I'm thinking at least 19 years old would be ok? And how could I epxlain how she doesn't age? Make her a ghost? Or make her some genetically engineered psychic girl that just stays young?

Make the game you want to make, write the story you want to tell.

Disturbing as it may be, there are games (and movies and books) that explore youth as murders and youth as victims. I mean, look at Lord of the Flies which is mandatory reading in many places.

In writing your horror script, if the script is better by being 17 or 19, do it.  Do you need an explanation about not aging? If so, why? I would consider the reasons she doesn't age are related to the reasons behind her other powers.

I would assume with psychic powers and with telekinesis, either she is ageless because of use of those powers, or she ageless for the same reason she gained the powers.  If she was abducted by aliens who gave her the powers, that's also why she is ageless. If she is possessed by demons to get the powers, the demons also make her ageless. Maybe assert that she became infinitely prolonged after an accident involving a few rubber bands, a liquid lunch, and a particle accelerator, that worked for Douglas Adams.

You're right there are many books and movies out there with youths being harmed or murdered. I just worry because it would be like a fighting game where the player can control how hard they want to hit or what deadly fighting moves they want to perform on their opponent and in this case if it's a little girl getting beat up, knocked out or killed, I worry that it would gain more attention than say when Mortal Kombat got criticized for having too much blood for kids to play.

If she's a little evil girl, she would have a huge brute support her fighting while she uses her mind powers from far away. i think I could work on the idea of aliens or demons giving her powers so she doesn't age, it sounds really cool! Or she was created in a lab with gifted powers? Does that sound overdone? And does little psychic girl and big brute sound like an overused stereotype trope?

Why does she have to not age? is the question I think, the reason why doesn't matter as much. No matter what you can't avoid critisism all together. Violence against a 15-16 year old girl compared to a 19 year old won't make a difference.

[Short form:it's a video game, and will come down to visual style]

Revan, it may inform your thinking to consider the transition from books to visual media, and what the effects will be. 

Two examples involving kids and combat:

Ender's Game. There is a killing early on that stays in the back of the reader's mind throughout, but remains in the background. In a filmed version, if the kids were as young as in the book, that scene would overwhelm the movie.

The Hunger Games: The books are arguably half horror, half political allegory. The movies for understandable box office reasons turned them into video games, taking away much of the blood and suffering (I'll table the art vs. commerce discussion). The older kids were not only played by young looking adults, but viewing audiences accepted they were older teens physically--something that would contradict the reality of food deprived physical development.