That would be nice, but not quite...
1) Every "legal" gun sale goes through background check. That means no buying a gun and then giving it to your idiot brother (if you have an idiot brother, you get the idea). You also have the waiting period in case somebody needs to cool off. I can't speak for the world, but I myself wouldn't know where to obtain an illegal firearm. Most people I would think would first start with a gun store/gun show/etc.
2) All firearms dealers, in the process of obtaining and maintaining a license to sell their product, need to go through a training class held by local law enforcement so that they know some of the signs to look out for and who they should contact if they suspect a problem. Much like bar owners need to train their staff who to serve, which ID's to accept, and when to cut people off or they get sued to oblivion if somebody gets drunk at their place and then injures somebody driving.
3) On the mental health issue, any direct relative can make a request to the police (parent, kid, sibling, spouse) that they visit the person in question and if they determine its a problem can request said individual avail themselves of the one free mental health exam allowed by my new law. Come in and fill out a form as I want to weed out hoaxes as best I can. No you can't request this on your neighbor because his dog keeps peeing on your lawn. The only exception to the direct relative rule outside of medical or law enforecment professionals would be: School principals/headmasters working with their teachers for a problematic kid in their class.
Again, this won't solve every case, but if there are any warning signs or an attempt to arm themselves to the teeth this could be the best way for everyone without enacting really any restrictive anti-gun laws.
EDIT: Missed your last question. I don't know enough about the CT horror to say. Columbine those kids seemed to be giving out enough signs to warrant intervention by lets say the school making a request.
