It's not random at all. He researches everyone, and chooses people he believes are wasting their lives, or have some flaw that needs to be fixed. People who are hurting themselves or others with their bad morals or habits, are given a chance to learn what they are doing wrong and turn their life around. In the end, they make the choice to live or die, and Jigsaw is just helping by creating a situation to make them realize.

Just go watch the movie again.

I have the sneaky suspicion it wouldn't make a difference...well, maybe in 5-10 years watching it again;)

Well, the point of the matter is that the government in the movies of Saw feel that he is a serial killer. Their reasoning is that he is inventing the traps and kidnapping the individuals so that they are placed in the intricate traps. Also if outside help is rendered the traps tend to go off. I know that it is to prove that the individual is the one that has to solve their own problems, but he is still considered a serial killer. And he has to think how this would possibly kill the individual, the thing he gets out of it is that he gets a clear conscious. He did give them a chance to walk away from it. Not unharmed mine you.

So for the 12 year old, yes he is a serial killer, but his hands only make the traps and he incorporates someone else to help kidnap, that makes him a kidnapper and a murderer. Because killing people with traps or cars or machetes is viewed in our society as still killing people. :)