I know what happened might erode your self confidence
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To fix a marriage especially after what you went through it needs both of you to work on not only you He should not tell you what you should do he needs to give you the enough time to heal from the inside of you he needs to understand your feelings and how hard it is the decision to move on that you decided to do he needs to understand it takes seconds to break a mirror but how long it takes to fix it or make it
About the vows you don't have to do anything you are not ready for it first i would suggest counseling to help both of you
and let him get things in the hard way so he will think more than twice before he do it again
As for the emails you know what you should do with them if i had them i keep them as evidence i keep them without checking them every time because they will increase my anger and keep my wound open but i might get back to them after everything being settled down and after i get over things i keep them to remind me about things i should not forget with time and keeping me aware of any change
About giving you his passwords it's a way he may used it to regain your trust it doesn't say he is not going to do it again or having another account but it's helpful to give you his willing and promise to be faithful this is a step it needs to be followed with so many other steps and strong patience because it's about time you may build this trust back or you may not as i said it's the hard and long term work for both of you to be able to succeed
I talked a lot but hopefully i was able to say things in a helpful way
Good luck