I am a little confused -- is he actually flirting/fooling around with other women? I'm not sure from reading these posts if he is untrustworthy or if this is an insecurity kind of thing.

If I understand this correctly, he is flirting via text with other women but telling you that nothing happened and claiming that you just have to trust him. He's making it sound like it's your trust problem, not his untrustworthiness. That's a sign, to me, to run for the hills. Telling you that he turns down lots of offers for no-strings sex is no way to make you have confidence in him, either. He shouldn't be getting those kinds of offers if he is letting people know he isn't available, you know?

What made you look at the phone in the first place?