It can get better. You're husband has completley disrespected you with his dishonesty and that is the biggest hurdle for you both to overcome now. Yes you broke trust in going through his e-mails butthat does not remove the crime so to speak. Building that trust is going to take a long time I would imagine. Setting boundaries and actually keeping them is the only way I can see to move forward. But he has to be honest and he is historically better at an affair aproach to relationships then an open communication one.
I think he is lucky to still have you in his life..escpecially after ditching his responsibilities as a father on your child's birthday. Parents that don't prioritze thier children above all else make my blood boil. Sometimes doing what is right for them means doing things that aren't easy on them..like divorce or seperation..but it is definitely not sitting in a parking lot chatting to some woman during thier birthday.