I would say no. I definitely wouldn't let my 13 year old brother play. I play WoW, and I don't think a kid that young can handle it. But I've met plenty of 17+ people who handle it well. I don't think violence is really the problem with WoW.
One, the social aspect of it is a little intimidating. To really have fun and do what the game is meant for at max level, one must find a guild and become a member of a group. One must be able to deal with sexism, racism, cursing, and downright disgusting conversations when playing with these people. Ok, not everyone is like that, but believe me, there are plenty of crazies out there. If you are 12 years old, you will be called out on that and constantly harassed. (Believe me, I am female and been playing for five years... I still get harassed every time a new person finds out I am female.)
Two, once you do quests all the way to level 85, you are left to raid (kill big bad bosses w/ others) or do player-vs-player. Either one usually takes time. When my old guild raided, we would raid for 4 hours (sometimes more), maybe with one or two 5 minute breaks. When my buddy does arena PvP matches, he and his partner play for upwards of 3-4 hours a day (sometimes every other day). (Agreeing with Paul on this one, you get way too involved in what you are doing.)
While some (like myself) find WoW to be an enjoyable way to pass the time, I don't think a 12 year old would handle the game well. Sorry if that was a bit wordy. :P