in my opinion: you will need to test the waters. if he wants you back as much as he says he does he should be the one to move to your city (but not in with you) to “date” you again. take the time to get to know each other again as the both of you have changed. you both did wrong in the past and both have had time to grow. i think your children will benefit from having both of their parents in the same home as long as his abusive behaviors is truly behind him. find out what he has done to work on himself regarding his behavior. the question is will the two of you really be able to leave the past in the past and work harmoniously towards a better future for the “family” unit. only time will tell whether or the both of you have changed (you had issues with infidelity and him with lying). you question whether or not you should believe him this time around but what about him trusting you with being faithful? you both have to learn to trust one another again. the only way to do that is to test the waters.
i don’t think we should ever forget when we forgive. we take our lessons learned and know what to look for and work on going forward. we sometimes need to forgive ourselves too.
