[continued from above]
I reckon that most of her kids' problems stemmed from their mother's example. They learned from her that it is okay to throw a crazy fit for no reason. Your teachings gave them a different example, and, now they have seen adults getting along together without the outbursts, so they know it's possible, and they can see that it's worth it -- more worth it than any tantrum. They're now choosing the better example, so good for them.
I think it may be that it is (a lot?) harder to reason with her during a tantrum. As hard as it is, you and your husband both might want to say, "I see that you are very upset right now. Let's talk about this again when you're feeling calmer." And then, look for ways to disengage from her and leave her alone for a "time out." When she's ready to be reasoned with, then you come to her (hubby doesn't have to just yet), hear her out as she vents to you about what was upsetting her, sympathize with words such as, "I can see why that would hurt you," and then when she's calmed down a little more, gently say, "You know that you were taking things to far in how dramatic your reaction was, right?" If she agrees, you then have an opening to say, "Would you be willing to offer our husband an apology? After that, the three of us can sit down and talk about what we can all do differently in the future."
The idea here is that whenever she throws a fit, don't just put out the fire and then say, "Whew, I hope that doesn't happen again soon." Instead, make a point to re-engage her as soon as she has calmed down, and reason with her. Try to get her to put herself in hubby's shoes, and feel the hurt that her fit caused him. Try to get her to remember what really set her off, and to say to your husband, "I still need us to talk about this thing that made me upset, but I apologize for turning it into an attack." Something to that effect.
Lurking under all that anger may be some legitimate concerns, but they get flooded out of the picture when she tries to address them by exploding. She needs to know that she can have legitimate concerns that deserve to be heard, without thinking that fit-throwing is a healthy or even effective way to express them. She knows this during her calm moments. Try to use those calm moments to negotiate an apology from her. The reward for the apology is that the three of you then talk (calmly) about what was setting her off, so that she will feel like she can be heard and understood without pitching a fit.
You may need to prep hubby, too, before the three of you sit down to hear her apology, to encourage him to forgive and reciprocate any improvement in her behavior. No saying, "You're sorry? You should be sorry!" Just expressions of appreciation for the fact that she's trying. Remind him that it's harder to be her than it seems, that she struggles against stress and buried panic much (most?) of the time and that the struggle does her in at times. For some reason, she uses your husband as a punching bag when she can't take it anymore, and that's not okay. But we can still feel bad for her, imagine how unhappy she must so often feel, and try our best to thank her for even the littlest (what looks "little" to us) efforts.
Re: the tie-dye business ... she thought it was private in an almost sacred way. You didn't see how it was private because she had already talked to you about it. I tend to agree that your perception was the more accurate, but once she has calmed down, you need to approach her gently, and see her side as much as possible, even if her side is skewed. It would help if your husband could say to her, "I didn't realize that you didn't want me to talk about the tie-dye business, but I see now that you trusted me to let you tell Lynn about it in your own way and time. I didn't mean to hurt or betray you, and I'm sorry if I did. Can the three of us sit down sometime and talk about ways to make the tie-dye business a reality? That would mean a lot to me." Yes; she needs to learn to apologize to your husband, and he needs to learn to apologize to her too. Even if it seems like she doesn't deserve an apology. She probably still needs it.
Your husband also needs bigger apologies from her, for the really big things she has done that hurt him in a big (and long-lasting) way. Sadly, it may take her a lot of practice at apologizing for "little things" before she's ready to apologize for the really big things. This is going to be a slow process. She and he both need to learn to put themselves in each other's shoes, to get better at doing so, and to do it often. I have a feeling they're not going to get there quickly or easily. Your husband probably feels like it's not worth the bother to try at all, because hey, she'll never treat him better anyway, right? It's going to take him some time to build up enough faith in her to try a little for her.
Re: the fling with her boss ... this sucks, but I'd try to let that one go. First of all, if even in her calm moments she won't admit to more than a few dinners, then there's no solid ground you can use for discussing it further. You'd have to get that DNA test, and you've kind of decided not to. You've kind of decided it doesn't matter too much who the biological father is; your husband is her daughter's father at heart. To me this suggests that it doesn't matter too much whether she had that affair either. After all these years, it's water under the bridge, and her boss is long since out of the picture anyway.
I would try to forgive and let that one go, with or without an apology. That means encouraging your husband to forgive and let it go too. It's too a heavy grudge to bear, it's like carrying a big rock around. It seems to me that if your husband could drop the DNA testing, he could probably drop that rock as well. I mean since we don't know who fathered the child, we can't know whether she had the affair, either. I think it's possible it was just a couple of dinners, but in any case, all the suspicion and uncertainty is just a weight and a poison. I personally wouldn't ask her to apologize for that one. I'd just try to let it go.
Eventually, when she's ready to, she should apologize for each time she ran off with the kids, with an especially contrite apologize for the time when she prevented him from seeing the kids for three years. Those years can never be brought back, so it deserves a very heavy apology, and it will still take your husband a lot of emotional energy to forgive. But that apology, and that forgiveness, are probably both far down the road (and I don't know they'll ever happen but I hope they will).
Eventually, when she's ready to, she should apologize for lying about him and his character, and for constantly exploding on him for five years before she got some medication for her condition. These are things she has admitted to doing, so she should be willing to apologize for them eventually.
I agree that he will probably never trust her 100%. But the three of you can try to work together to find out what she can do to increase his trust just 1% at a time. Even if his trust level is at 0%, there must be little things she can do to raise it up to 1%. She's taking her meds, and she's treating him better; that's a start.
It's all about discovering the little things that can be done to improve the situation just a little at a time. Even if we've combed through 100 little things we're already trying, if we can tease out just one more new little thing, then our effort has not been in vain. Remember, it was a flaw one little O-ring that caused the Space Shuttle Challenger disaster. You never know what easily overlooked detail could end up making all the difference in the world. So, let's keep combing through the wreckage, and try to find the one O-ring that can make the next shuttle fly. I can be very patient in this type of project, and I know that you can too. The dream is worth it.
Love means never having to say, "Put down that meat cleaver!"