Quick speculation. As 13 year old kids like this usually create an identity from multiple sources- so far we're pretty sure the main movie was Oldboy-- I wonder if he didn't also identify with the Question (the comic). He signed his name as a question mark, he wore the same baseball cap, gloves with short sleeve shirt. And, in case you weren't once a lost male adolescent, the Question is Vic Sage, ex-philosophy major turned TV reporter who is also a vigilante. And the 30+ issues are all about his struggle with identity.
None of this is relevant to why he did it, of course. It's fun to speculate, but of no predictive value. These are costumes, roles. They just help him "get into character," which in this case is a man alone fighting injustice, greed, and abuse, using whatever tools he has. Of course, he was fighting college kids, and his tools were semi-automatics, but who am I to dispel a good illusion?
Finally, let me reiterate that the presence or absence of mental illness is irrelevant, at least as far as prediction goes. So let's say he was a violent schizophrenic. Would you jail him before he commits a crime, just in case? Bush does that with Guantanamo, you know. Do you send him to a psych hospital, like they do/did in the USSR? That process will be magnificently abused by your government, I assure you.