I'll admit it, I'm a terrible person. Right?

My boyfriend of 7 months & I only have one slightly complex disagreement: He still emails & texts his ex.

This I knew getting into the relationship & as a fairly confident, intelligent, self-proclaimed "non-jealous bitch", I wasn't too concerned about it. Just a side note, the two of them dated for three years and lived together. They broke up because of distance and differing goals (he is getting his PhD [as am I] and hopes to travel the world, she wants to be a homemaker).

The problem began when he used my laptop to check his email...and forgot to log out. I'm sure you can see where this is going. I took my laptop home, realized his mistake,...and proceeded to read through their entire conversation thread. The first time this happened was a month ago. Even further, I noticed a post-it note above his desk with all of his passwords. His listed email password was "i love(Ex's name)1". Sickened, I tried it and it works.

It has become a problem, logically. I likely read his emails more than he reads his own. He also in person and in conversation has refused my request to tell his ex that he has a new girlfriend because he, "hates to hurt people" (Who doesn't?). I believe it's better for her to hear it from him than from friends.

In the emails she still confesses her love for him (and he does the same to her!), sends him dirty pictures, and updates him multiple times on the proceedings of her day. He does the same (minus the dirty pictures). The one good piece of information is that when she asks whether they'll ever get back together he always insists that they won't.

Problems have arisen when I ask him questions about certain things (often, admittedly, prompted by something I have read in his emails), and he responds with opposite opinions than those stated to his ex. For example: Me: So dear, did you talk to your ex at all on NYE (I was away on vacation with my family)? I know that she likes to drunk dial you." Him: "Of course not." His email from her: "Thank you for talking to me so much through NYE. Hearing you tell me you love me and fake kissing you at midnight made my night." He also has told me (threatened me, perhaps?) that he broke up with the girl he dated before me because she tried to get him to stop emailing his ex. He states the ex is still his best friend and he'll never stop. When I asked if I could meet her and become her friend too, he acted as if I was crazy.

This is ruining my attitude towards him and towards continuing our relationship. I know it is completely my fault for invading his privacy, but a part of me is glad that I did. To me, it shows that he is not completely committed to our relationship, and he has no qualms about lying to me (I know, I know, I'm lying to him by checking his email.). Am I completely wrong in continuing to check his email? What about my insistence on his telling his ex of our relationship? I think his relationship with his ex is ruining ours. What are your overall thoughts on my future actions?
I'll admit it, I'm a terrible person. Right?
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