I'm questioning the mental health of a very rude and incompetent employee
Do you think there might be some kind of family problems the employee is experiencing?
No! His family is nice and charming, however he himself is a bad apple and quite rude.
It is wrong of that employee to be so rude. You should probably tell your boss on him.
I've told him my theory of poor mental health and he dismissed it, saying that he had been there longer than me.
Well, since you already expressed your theory to your boss, you could try one more time though I wouldn't necessarily expect a different result. It might not be a mental health issue, just a personality trait. Professionally speak to your boss and back up your complaint with examples.