Trust can definitely be rebuilt if the person being dishonest is willing to do the work on their inner selves and also with you.
If DH does not want to do any of the above you may have to make some difficult choices. You cannot continue to let him put your health, mental and otherwise, at risk while he diddles with other people behind your back. Loving him is not enough and at the very least you've recognized it.
It'd be great if you could invite him to the forum.