No, they do not. Anyone can justify any belief regardless of faith. I will admit faith is an easy target to justify horrible things, but its not at all the only way to justify things like that.
Remember what "faith" means in this context: the conflation of personal belief with objective reality. The act of "Justify[ing] any belief" is an act of "faith".
That's just how people work. Instead of admitting their beliefs are wrong, they will do mental gymnastics to justify them.
That is how certain people work, not all people. You have identified a set of people who "conflate their personal beliefs with objective reality".
The underlying problem is absolutely bad mental health. Not necessarily a mental illness, but bad mental health in general.
I don't think so, but let's check on it: is it a mental health issue when we use an incorrect order of operations in a mathematical statement? For example, x=1+2*3. Is the person who gets "7" mentally healthy? Is the person who gets "9" mentally unhealthy? What of the 3-year-old, who has not yet been taught numbers, and scribbles a stick image of a cat on the sheet?
An individual who does not comprehend the meaning of PEMDAS/BEDMAS is still capable of rational thought. The lack of knowledge will lead them to a fallacious conclusion, but their process of reaching that conclusion is still rational.
A deliberate refusal to accept and follow PEMDAS/BEDMAS rules is an error, but is not an indication of mental illness.
The knowledge that individual belief must be subordinate to objective reality is a philosophical model that not everyone has learned, but ignorance of that philosophy is certainly not indicative of a mental health condition.