You should definitely forgive him. when you confronted him he told you the truth and even let's you go through his phone, phone records, emails, etc. He told you a little white lie because he didn't want you to get upset and it's obvious he doesn't want anything to do with his ex. he just wanted to tell her about the picture. I think he is being extremely generous letting you go through all his personal stuff and information. clearly he has nothing to hide and you should just let this little lie go. he sounds like a nice guy.
oh and don't take this the wrong way but I think you might be just a little crazy to go through all his personal stuff and have his passwords. if you trust him then you shouldn't have to be looking at all that stuff to make sure he's not talking to other girls or whatever