<human>: Can you condense the text into three sentences or less?

He is 23, I'm 20. He has cheated in the past but admitted to it, after a break I decided bygones are bygones, etc. 
I have the password to his email and we share a laptop. One day he left his email logged on, and I saw some lewd emails from him replying to craigslist ads. This sparked my curiosity, and I have been watching ever since. You see, he's a sneaky kind of person. He withholds the truth but swears he doesn't try to lie. All that.

A week ago, I confronted him about the situation. To be clear, I don't believe he is acting on these emails at this point. I told him that it is unacceptable to be asking for a "twitter girlfriend", telling other girls that they are cute and otherwise flirting, and speaking inappropriately to other women was unacceptable. After his continual denial, he finally admitted to it and said that it was just us from now on.

Now, today, his email had a message about him replying to a female wanting a threesome before 7 today. I don't know where he's been all day, he said a friends, but he's a liar, as we've discussed. I don't know that he acted on this, he better not have, but he said he was "interested" in the ad. He came home, I've been cranky and tired so I have said nothing yet. I napped. :P

So, my questions are as follows:

1. Am I acting out of the area of appropriateness by stalking his email?

2. How do I proceed in this discussion?

3. Is it even worth the discussion, or do you all think this is not about to last?

Thank you everyone for your help!
<bot>: sneaky bf talking inappropriately/sharing pictures on craigslist, talking to other women inappropriately, lying, etc. Need to know if I'm going crazy or if he's worth the effort.