Please ignore those saying he is absolutely having an affair, the reality is none of us know.
You've written the op because you clearly think he's guilty, in reality he's had a drink with her and works with her. That's it.
If my husband was reading my work emails, questioning me like this, and snooping to the extent you are, looking at history, deleted items etc I'd be friggen furious. He is also going through a horrible time at work and with his parent and now needs to deal with your behaviour on top of it.
You could be right, but equally you could be wrong and simply causing your marriage massive damage by the way you are treating him in this stressful time in his life.
You say you're happy and secure, but have previous, I don't think you were happy and secure, clearly you were constantly on the look out.
As said, he could be shagging her, which I think is unlikely, or he could simply be furious at the way you're treating him.
No one can guess, the only fact is if he is innocent you're doing your marriage irreparable harm. If he's not, you're not going to find out by behaving like this.