There's definitely a trust issue there, both with him being deceptive and with him being jealous (reading your massages?! god! to me that would be a good enough reason to get the hell out of there!)
Since I don't know the relationship I can't tell you to leave or stay, instinctively, I have a bad feeling about this, but if you feel like there are good thing to salvage there, then the two of you have to do a massive amount of work to make this a trusting relationship, starting with recognizing that the problem exist.