Gemmy, I read your story and was heart broken by it. I am so sorry you are going through this. We all say it and we all mean it. I can't tell you what to do, I can only give you my own thoughts and reactions which may or may not be useful or helpful.
First of all, I think anyone who cheats around a wedding - before, during (yeah that happens too!), honeymoon, within first year....is not committed to the marriage, or maybe any marriage. Some people are NOT marriage material. Not marriage material, not monogamy, not open relationship, not nothing...just not relationship material. To cheat at that time means there is no basic commitment to the marriage, it's being done for other reasons. Maybe because it's "time" to get married, pressure from family, desire to have children, the proposed spouse looks like a good Plan B if they can't find the Plan A they want, maybe it's a financial arrangement - whatever. But when you find out someone has cheated on you at that time - around the wedding, beginning of the marriage, it means they were never committed to the marriage in the first place. So do not worry about anything you did, or didn't do, or said, or what happened, or didn't happen, or how you felt, or how she felt or any of that - she was never into you in the first place, this is not your fault, this is her nature, and she was NEVER committed to the marriage. I think when you start off this way, I don't think it really gets any better. People pretend, they develop a persona, which is what I think she did, they get good at lying and covering up, but the real person - the core person....is never there. You don't know your wife, you only know what was presented to you. NOW you are getting to know your wife, and she is NOT marriage material for you or anyone else. SO DO NOT BLAME YOURSELF FOR ANYTHING - she IS the essential problem. She never loved you. I really want you to understand that....it's not your fault, it's not you, it's not anything you did or said EVER....it's from the start - it's all her. She decided you were acceptable for this purpose and went ahead with it and then did her supplementary real life on the side. The affairs are the REAL her. Always remember this.
I know a lot about mental illness and substance abuse, etc, I've seen a lot of it in my family, I grew up with it. The panic attacks are a controlling device. I'm not saying she does't have them, but it's how she uses them. Frankly I literally DO have panic attacks - it's a physical-psychological condition brought up by an entire childhood of abuse and neglect. So I have period when I get overwhelmed, usually by physical stimuli. I take medication for this. BUT I DON'T USE IT AS AN EXCUSE OR TO CONTROL OR BLAME OTHERS. She does. She uses this as a way to control you and the situation. I don't give a shit about her "panic attacks" even if they feel real to her because of how she's using it to control your reactions, your behavior, your emotions and how others deal with you. Don't let her control you - if she starts the panic attack bullshit, leave if you can or call an EMT but do not respond to it. Leave the room, if she does get bad, call an EMT. If she has to face an actual result for this bullshit, she'll stop. This is like suicide threats - people usually do this in relationships to control the other person, if they continue, I would call in the professionals.
You have a problem, I've seen in other relationships where the wife deliberately uses the spectre of domestic violent - fake claims of DV again to CONTROL THE RELATIONSHIP AND CONTROL THE PERSPECTIVES OF OTHERS. Your wife, IMO, is an absolute piece of shit. I HATE women who do this as they make it so hard for others. A friend of mine had a wife who cheated on him for a couple of years, and stole hundreds of thousands of dollars from marital funds, and then started telling him she couldn't be alone with him for discussions, etc, because she was "afraid". She was trying to set up the DV scenario bullshit and unfortunately the woman is almost always default believed (I often don't believe them personally cause I know how women lie) but it's the default because a man could seriously hurt or kill a much weaker woman....of course. To me the answer to this is always to protect yourself by recording what's going on between you when you have to be together. Depending on the law where you are, you have to let them know (some areas you don't - you can have 1 sided recording but be governed by your local laws), but I would always record transactions, write them down afterwards, document document. YOU CAN NEVER TRUST THIS WOMAN EVER AGAIN ABOUT ANYTHING, ANYTHING AT ALL. Everything should be documented and hopefully recorded. Don't ever trust her again about anything, I cannot stress this enough, and NO, she will not change and therapy won't matter. This is how she is and how she has been from the start, you just didn't challenge her before. Now that she's exposed...you're a threat that has to be contained.
I would not advise reconciliation of any type, what is there to reconcile? And she will always have the upper hand because she's a good liar, she can put on a good show, and she willing to create DV scenarios that will always endanger you. Again, talk to a good lawyer, your aim should be DIVORCE, and ask about how to do this safely. DO NOT RECONCILE FOR THE KIDS. This kids, as you've indicated know something is wrong and it sounds like at least 1 of them senses it's Mom. I would get out of the house, find another place and see the kids separately. Unfortunately this is the situation she has created and you don't know to what lengths she will go to remain dominant. You just don't. Assume the worst. Minimum contact directly with her, hopefully recorded, documented, or witnessed, is best.
Once you get out of the house, except for the kids go minimum contact with documentation as I say. Don't have sex with her again as she could use that against you and if she gets pregnant with a boyfriend she could say it's yours, etc. Get an STD test. Always assume she is lying, do not do any marriage counseling with her, do not accept overtures, alway assume the worst. She's never loved you from the start, you were a means to some end in her mind. I'm sorry but...that is the way it is.
As for the kids, when it is safe enough, probably when you no longer live in the house, tell them, age appropriate, the truth. Tell them Mom has boyfriends and has been doing this since you were married. If they want to know why, they have to ask her, but that's what's going on. She'll have some bullshit to tell them, but always tell your kids the truth. ALWAYS, in an age appropriate manner. They know something very bad is going on and they may even sense why. So don't lie, it undermines your credibility and your relationship with your kids.
As for your sister, I have to disagree with Bigger on this one. I'd cut her out. Not in a angry or attacking way, but just be aware that this person has known you her entire life and she still sided with your wife without even hearing out your story or giving a remote shit. She really thinks YOU are the type of person who would do DV? Then she is NOT someone you should ever trust or have anything to do with other than the bare minimal. Don't let people into your life once they have screwed you over.....she's shown you what she is and what she thinks of you. BELIEVE IT. Some women are so into feminist ideology that it trumps anything else including their relationships with their own male family members. You don't need this.
I'm sorry to seem so negative. It's the way I am because of what I've seen in life and most of the time....I'm right. I don't believe in unicorns, or fairies suddenly flying out people's asses, or the cheater suddenly SEE THE LIGHT! or any bullshit like that. I believe in protecting yourself, being honest and authentic, not allowing other people MORE opportunities to hurt you and admitting the truth about people and situations even if it hurts. It's best to call cancer what it is and treat ir properly than hope for the best. Find yourself a hard nosed divorce lawyer and work this out with him or her, the best plan for you and your kids. BUT DON'T STAY IN THIS SHIT SHOW....you're endangering yourself and your kids and you don't know how bad this can get. This woman will not improve with age.