To be perfectly honest, I think this is all a bit redundant. There are two terms used to describe illness, mentally ill and physically ill. The first describes an illness of the conscious brain, the second an illness of the body.
What if all the perpetrators were called to task? Who would we be then?
There is such a thing as just a bad person. There are lots of bad people... sure, some no doubt are mentally ill to some degree from their own lives, some are just bad people and they get a rush, feel excited, doing bad shit.
I believe the statistics for traumatic stress based disorders within US prison populations is above 50%.
Not everything in this world has a hidden agenda. Some people are just good, some people are just bad.
The more common reasons people do bad things, is because they don't have something, want it, have to power to take it, and make a snap decision to use that power to take something that isn't theirs. There is a reason why violence is so high in poorer populous than medium class populous. Many see others having things, things they worked for and earned... the not haves want it, so they choose to do bad and take it, regardless of the other person.
Culture has more to do with bad things happening than mental health. You're barking up the wrong tree and need to research that topic, and you will find culture is the leading cause of issues, not mental health.
I do understand one may think its as easy as labelling mental health on anyone who does something bad... but it just isn't the case, and has far more to do with demographics and cultural upbringing than anything else.