I'm no parent, but I have gamed since I were 6. Personally, I think it depends on the kid wether it hurts to play certain games or not. Somewhat smart kids know that GTA (which I think is the most relevant game) is fiction, it's just a game, and the less smart know that even if they would want to do anything like that in reality, they would get caught. I honestly think the hip-hop culture is more "damaging" then playing games.
Except GTA, I can't really see a game damage a kid in that way. I guess there are games that could make them have nightmares, and that depends on what the kid is afraid of, ofcourse.
Now, I'll try to answer your questions more directly.
First, if she wants to get/play a certain game, and you say it's not okay, I think she'll go to a friend. That's what I would do, and (honestly) everyone I know. I don't think games will make her stay inside more. If she is a stay-inside kid, she is - it doesn't matter if she likes games or not, in my opinion. I'm extremely nerdy and only go outside when I have to, but I have nerdy friends (less nerdy then me, I guess) that play a lot of games, but still spend quite a lot of time outside. If you avoid saying "No, why don't you play some DDR instead of going outside", or anything like that, I don't think you'll 'teach' her to stay inside.
Oh, right, if she wants to play a grown up game. Well, this depends if you mean.. Bar games, or violent games. Bar games, I would wait until I'd let her play that. Horror depends on the kid, as I said. Violent? Well, I played GTA and everything like that, and I'm the most nonviolent person I know. I haven't hit anyone. Ever. Not kidding. Can't help you making that decision, all I know is, that if she wants to play it, she'll probably go to a friend or something.
Sorry if it's hard to read my post or something like that. English isn't my first language, and long posts tend to be one big clutter if I produce them. (Oh yeah, games made me learn english. Really don't think that matters to you though ;D)