Hi. I'm sorry to hear about this. It is hard for us to say what is causing his behavior. You think it is new? Or is this just who he is? Anxiety can present itself like anger and is a common pattern. Quit, hot head reactions are often present in people with anxiety--- agitation is a symptom. Could you all encourage him to discuss this subject with his doctor?
Regarding the lying, I would not say that comes from stress from lack of sleep. If he's a pathological liar, he is a pathological liar. All you can do is protect yourself, emotionally and other ways. I hope you and your mother can get out from under soon.
He might not be getting any sleep. It can make people very stressed, and unable to cope with emotions that an average person could blow off. I'm sorry he's being this way, and am glad you can separate the bad from the person, but also don't put yourself into the line of fire and don't let this be a pattern you would accept in your own personal relationships when you decide to get married. You deserve better than this and walking on eggshells is NOT acceptable. I hope your mother and yourself manage to move away if he cannot find a way to get more rest or other help for his issues.