Here is my personal opinion - no matter how messy/slobby/lazy you were as a wife/mate/partner, that is no reason to cheat. It just isn't. Cheating is by far one of the most painful things you can do to someone. Those two things just don't compare.
Now, put the shoe on the other foot...Had HE been working 60+ hours per week, would you have stepped in to do more? How much free time did he have versus you? I know when my H worked a bunch of overtime last winter, and I had a week off, I took over everything. I didn't become resentful or fall out of love or fall into love with someone else because of it. H does a little bit more than me because he is crazy OCD about things being spotless and clean 100% of the time. He has assured me a zillion times he is not resentful because of it. We've talked a lot about this issue because I do more stuff with the kids and feel it winds up evening out.
I guess what my point is that I don't think you should beat yourself up or take the blame for his actions. I definately think becoming cleaner and a better cook are things that you can do to improve yourself and let him see those changes as positives, but not as an excuse for his behavior.
As far as the going with him, I really don't know. It seems like a huge step in the right direction in terms of him inviting you. I do wonder what OW would think if she knew you two were going to be hanging out at his family's??? I would think that would be extremely upsetting to her if she knew...I am so evil though I would probably go and then make sure it got back to her so she would fight with him on it...LOL But I am a meanie like that! You seem really nice!