Sorry OP but he sounds like a controlling arse.
Kicking off if you get complimented is not ok
Telling you that you have to cut all contact with ex's who remained friends is not ok
Set in his ways sounds like an excuse not to compromise and in a partnership that is not ok.
What do you mean by old school? Does he expect you to be subservient and not challenge him?
It sounds as if he is engaged in some kind of emotional affair or about to. You don't generally flirt with friends in messages that you keep secret from your wife. What made you read his messages in the first place?
Also what context do you mean flirty messages? Is it innuendos and compliments or is it more directly sexual?
If you do have the conversion with him be prepared for the usual crap script:
He will be outraged that you have looked at his phone and try to make the dominate the entire conversion about him being the victim as you have violated his privacy.
He will minimise the content of the messages and call them banter and tell you that you are overeating.
He will tell you that he is allowed to have friends and try to make you feel bad as he is lonely. He may even accuse you of trying to isolate him from his friends and try to make you feel like the controlling one.
He may use gaslighting strategies you until you start to question you own rational thoughts and feelings.