I could not and would not live with someone's mistrust and snooping in my personal information, so I can understand your irritation and anger.
But he will do what he will do and the only response is to change yourself and your response to his lies and other issues.
What things do you need to have as "givens" to live in peace with this person in your home?
Are you unwilling to live with lies and active drinking?
Are you unwilling to live with invasion of privacy?
Are you unwilling to accept constant criticism of your actions?
What consequences are you willing to enforce if these are unacceptable to you and he keeps doing them?
As SK says, it isn't so much about being "right" as finding peace for yourself.
What things need to happen for you to find / create it in this situation?