I have a quick question.
I am worried about my responsibilities as a step-mom, the legal side (for example, kid goes outside to play and gets hurt - am I to blame I was not there? or in case of anything serious - can their mother sue me?).
Are there any laws defining what exactly I am responsible for?
Where to find and read?
I don't think you're any more liable than a parent would be, or anyone else watching someone else's child. I mean, technically speaking, if one of my dds' friends were to get injured while playing in our home or in our yard, that parent could potentially sue us to cover any medical costs. I'm not sure how medical issues are handled with your dh and his ex. Presumably she wouldn't need to sue you as he's probably paying child support, and may even be covering his child on his own insurance policy, and would probably help out with any medical bills over and above what was covered by the insurance.
Back to the example of one of my dds' friends being injured while in my care, that parent could also call Child Protective Services if he or she felt that the injury had occurred due to me not adequately supervising. Especially since I have my own children, CPS could become concerned if it seemed that I was grossly inattentive as what happened to the other parents' child could theoretically happen to my own children.
I don't think most parents would do this, as involving CPS could complicate their own lives as well.
In step parent situations, it seems more iffy. If your dh's ex is not happy with the current custody arrangement, she could be looking for ways to have you guys declared unfit. Has she given you any reason to assume she's unhappy?
I just noticed that you are worried about your step child being injured when you are not there -- do you mean when your stepchild is in your home and being supervised by his or her own dad while you are out?
I don't see how anyone could pursue you personally in this case, although, if you have children of your own in the home and your dh's ex decided to call CPS on him because she felt he wasn't adequately supervising their child and an injury occurred, this could certainly affect you as well if an investigation were opened and a caseworker was coming into your home to check things out. And medical costs could presumably affect both you and your dh, depending on how it was handled. For instance, if you own your own home and your home insurance premium went up, this would affect both of you. Or if your dh had to pay a large bill, this would probably affect you, too.