It really depends on how mature you think your child is.
My younger brother has been playing the Call of Duty franchise since he was 12, I presume my mother assumed he was mature enough to handle the content in the game.
Call of Duty does have some violence, it is a war game after all. Whether Black Ops is extreme depends on your definition but I wouldn't say the violence is senseless - in fact the game itself is quite educational in terms of it's representation of war.
There are some scenes which may not be neccessary and are quite extreme for children.
I've completed the game myself and it didn't contain anything you wouldn't find in a general horror (I'm a 20 year old female.)
Perhaps 14-15 would be a usual age to give a game like this to a child if you think they can handle it.
As for the online multiplayer, your child is able to play and chat with random strangers. There is no restriction on foul language (although there is supposed to be) so that may be a concern to a parent, but the majority of the time people choose to chat with their friends rather than random strangers.
In multiplayer, there is no storyline. There is different modes (such as Deathmatch) where you just have to shoot the opposition. So there is no sexual content there.
I can't remember the main game containing sexual content. There may be sexual references but I sincerely doubt it. I seem to remember some scantily clad females in there at some point, but nothing you wouldn't see in a music video.
Honestly, it depends on the kid. I'd give it to a 12 year old if I deem them mature enough, this isn't going to mentally disturb them and even though the violence is extreme it's presented in context. If that makes any sense.
And when he says all his friends have it - he won't be lying. It's one of the fastest/best-selling games of this year.