Before you worry to much about the forgiving part and who you want to be with. How about you focus on the fact that your W is mentally sick. The mother of your children is sick. I'm not saying be her rescuer at all, but I think strength and mercy needs to be shown here. Again I suggest you start researching this disorder quickly.
All of her actions could be because of a severe manic episodes. Outside of her control and definitely yours. This marriage will not be salvagable unless she gets help first.
You do have some tough choices 9, but I think your children and W deserve your attention for right now. Outside of this, the OW is a distraction to you. How can you even begin to worry about a relationship with someone else with all the emotions YOU are going through right now?
Brace yourself, because the little bit I've read on bipolar it could turn into a 180 degree wild ride again.
Keep focused on your family and let the dust settle a bit. Hang in there bud.