OC, I'm sorry you are here brother. Your situation and your wife's actions are just unthinkable and my heart aches for you and your children.
You've been given a lot of good advice, some of the most important being that this is not your fault, and you are not alone.
The only two pieces of advice that I dont agree with is the recommendation not to talk with your Bishop and the suggestion to serve your wife divorce papers publicly at church. My perspective comes as a fellow active member of the church who has been on the recieving end of infidelity/betrayal that nearly cost me my marriage and my kids intact home life.
The primary reason I reccommend talking with your Bishop is that you and your children are going to need as much support as possible to get through this, and men tend to isolate themselves when they have been betrayed in this manner. You've done nothing wrong, and you shouldnt be cut off from the love and support you are going to need to get through this.
The other reason is making sure you control the narrative before her version of "the truth" comes out. From what you've shared I'm guessing (correct /ignore if I'm wrong) that your wife has the reputation of a kind, loving, selfless woman who takes in troubled youth and raises them as her own. The type of woman who is incapable of the things that you now have proof of.
Therein lies your peril if you dont get out ahead of this. When you expose her, she is going to come to the harsh reality that she may lose EVERYTHING. Her marriage, financial stability, her good reputation, her membership in the church, her relationship with her parents/family/friends, and possibly her home and children, not to mention her new lover (as sickening as that is).
She is going to be abruptly backed into a corner. This kid will be in the same boat, and you dont know how either will react.
When wayward spouses get confronted, it is very typical for them to lash out as a self defense mechanism. I've seen first hand two instances in the church within my very close circle of family/friends where the husband was accused of domestic abuse and inappropriate actions with their children during divorce/infidelity. And while nobody knows with 100% certainty what goes on behind closed doors, I know from first hand experience that the two accusers were manipulative people prone to lying and the actions seemed out of character for the accused husbands. In both cases, the husband lost complete access to their children (not even supervised visitation), were financially ruined, and one was so devastated he ended his life a couple years later. You and your children cannot afford for this to become your reality.
In your situation, until you confront your wife and ascertain how she reacts, you have to assume that you have 2 enemies living in your house conspiring against you. In fact, you already know they are betraying and conspiring against you.
It may not happen, but if your wife decides to make false allegations, she may enlist this kid to back up her story, and then its the word of 2 against 1, and she could go to the Bishop with her sob story of your abuse, and could tell the same to anyone else at church who will listen. I think it's human nature to give more weight to the first version of a story you are told, and if your wife is viewed as a saint incapable of such actions, your side of the story may not be believed. I know very few men in the church or otherwise who are respected enough to be given the complete benefit of the doubt in that situation. Fortunately, you have proof of her/their actions. Guard it carefully.
Your wife and this kid may have abandoned every tenet of their faith, but that doesnt mean that you have to. Your Bishop absolutely should be a trusted spiritual advisor that can help support you and your children. That said, your Bishop is not your lawyer, is not your therapist, and unless he has been similarly betrayed lacks a deep understanding of the trauma you are facing. That doesn't mean he cant still support you and discreetly enlist the help of others. And dont forget, you still have agency to follow any counsel he gives, but there is no commandment for you to do so.
I agree with others that you need to confront this situation soon, but not before you consult with a lawyer about your rights in getting this kid out of your house, how to handle the complex living arrangements between your wife and in laws, any paternity related issues, and any other financial protections you need to get in place to protect yourself and your kids.
If you choose to talk to your Bishop, I think there are some timing considerations. If it was me, I would talk to him no more than 24 hours before I confront her, and have him on standby for afterwards to put into action any support you/your kids need.
In regard to serving your wife divorce papers at church, while it might feel good to think about blowing up her reputation, that would look very vindictive on your part, and could have negative impacts on you and your kids.
Also, once your ducks are in a row and you are ready to confront her, if you have parents or close family nearby, I suggest having them watch your kids for a few days. I think for all involved, the AP needs to be out of the house and away from your children during the confrontation. You dont need that wildcard in play, and if there are any firearms in the house that this kid could gain access to remove them.
Again, I'm sorry you are in this situation, and I pray that you and your kids get through this with as much love and support as possible