Every day things like kid or state of being are a regular part of what I talk about with people I have daily contact with.

While he may have been turning the phone away from you while texting, he was texting in your presence and when pressed, allowed you to view the conversational history. I'd believe him when he says he hasn't / isn't having an affair (i.e. sexual relationship) with her.

I understand your concern about how this may effect the future. You'll need to determine what you can and cannot accept, communicate that to him, recognize he is a grown man who is going to make his own decisions and act accordingly.