If you let someone use your computer to check e-mail or get on FB, you are not giving them permission to go through your papers on your desktop or in your files. This would be like going over to someone’s house and while they’re in the kitchen fixing you a drink, looking through all the drawers in the room or going through the stack of mail on the desk. Even if it’s a huge mound of mail, it’s just not done. And certainly not on a third date.
I think you need to confront him about it, see if he lies to you or not, and make your decision about the 4th date from there. If you do, you’ve learned something about his impulsive nature, upbringing, and sense of boundaries. And that you have a fundamental difference.