This is a really horrible sitution, I'm very sorry you're in it, I think yur husband has handled this int he worst possible way. He moved out just so his kids wouldn't have to do chores? That's utterly ridiculous. Why on Earth did he move back in? That many transitions is likely difficult and confusing for his children, making this even harder, and it sends horrible messages about his relationship with you.
I've seen some parents in such a situation who just have to have separate rules for their children (so mom says her kids have to do their chores and be responsible, dad says his kids don't). I've always seen this in situations where one parent's kids live there full time (and so have to do chores) and the other parent's kids go back and forth between houses (so that parent wants to be the 'fun' parent), this would likely be easier if you had your step-kids full time. It's not a good situation at all, it's likely to lead to resentment between the step-siblings, but it may be what you have to do. Ideally, you and your husband would be co-parents who agree on how to raise all of your children both bio and step. If that's not possible, ultimately your children are yours and his children are his and if you can't agree on a way to parent all of them harmoniously, you have to do what's right for your children.
Although it's hurtful to you that they aren't bonding with you, it doesn't sound like they're bullying. I know it's not ideal, but it is okay to just let everyone co-exist without properly forming family bonds. Do your step-kids have their own room(s), or do either of them have to share with your kids? If they have their own rooms, then you can let them get as messy as they please and you can leave it to your husband to set or not set those boundaries- just close the door so you don't have to see it. If they share with your kids, then the refusal to do chores is much harder.
I would also give his daughter some slack, depending on her overall attitude and exactly what was said. Even bio-kids do that. Can you honestly say that you've never talked to a friend and vented about your parents when you were upset with them? I've seen plenty of people use these forums to complain about partners, parents, in-laws, children, etc- it's human nature to vent, and most of us try to do it in a place that no one gets hurt, which your daughter wasn't successful enough. It's hurtful to hear mean things said about you and your child, I know, but it could happen even if she were the most bonded, loving, respectful step-daughter ever.
Also, it sounds like your methods of trying to connect with them may be based on you pushing your own interests. They like video games? Okay, how often do you sit down and play video games with them? Have you looked up the instructions and walk throughs on their favorites so you can keep up (unless they like watching newb mistakes- some do)? Do you ever look into video games that you think all three of you would enjoy playing together? If you just don't think they should play video games all day- again, I think right now you need to let your husband parent his kids and focus on trying to bond with them. Meet them at their level, not on the level you want them to be.
Family counseling may be a good idea. It seems like everyone is on different pages, and it also may help the kids sort out their feelings.
Because your husband has already decided to move out once- you may want to look into Living Apart Together
. I know some people who prefer this, and it may be what your relationship needs until the kids are grown. It won't be as easy for you when it's not what you want, but it is certainly possible to have a long term, committed, monogamous marriage and live apart. Again, I'm mostly suggesting this because your husband has already been willing to move out to please his kids- it could happen again.