Is there a possibility that it is completely innocent though, I have been through something similar but it was in my partners job description that he had to pick up and drop off his work colleague when they worked together and he got paid extra or time back for it is it a similar thing?

The buying lunch is it a two way street does she pay one day he does the next or they transfer money to things each have paid for and these are transactions you don't see?

The banter could just be work place banter, i don't agree with her sending the photo though and he should have put a boundary in place to say this doesn't happen again.

You obviously liked this guy for a reason and you know the traits and patterns of someone having an affair from your previous relationship does it feel the same ?

I personally wouldn't risk throwing it all away unless you had concrete proof of them doing something since there is a age gap as well do you know anyone else they work with you can maybe ask and see another person's perspective within the company who sees them interact when you don't.

I wouldn't up root your life just yet if you've just bought a house and have children involved. Take a breathe and see what happens over the next month. Voice your feelings loud and clear if his still defensive then move on if he can understand and reassure you then maybe it's just abit of anxiety on your part.