Does Shahrazad get involved in the stories she tells, as an active narrator? If not, why not?

I'm not sure what you mean by "does she get involved?" Shahrazad both narrates and acts as storyteller. Let's face it, her life depends upon actively keeping her husband (King Shahryar) attentive and enthralled. Thus, in order to keep him focused on her story, she needs to draw him in..... I would consider that getting involved.