Will parents be held accountable for the video games their children play? What if one is a step-parent?
Whether a child genetically comes from you or not, the fact that you have taken on the role of a parent means you are responsible for the raising of the child into an adult. That responsibility does not change whether you are the biological parent, step-parent, adoptive parent, or guardian. In your case, you knowingly shouldered the burden when you said "I do" to your spouse.
Children don't often see the ramifications of what they do. That is where a parent has to step in and provide reasonable boundaries so that the child doesn't do lasting harm to himself or others. "Fathers, do not provoke your children to anger, but bring them up in the discipline and instruction of the Lord" (Ephesians 6:4).
In regards to suitable video games, see some of the prior discussion: