Can your questions possibly be answered without considering: "Based on what Bible can he tell you?"
I WISH somebody would have stepped up to the plate and told my husband to repent for his evil against his family when he carried on with his mistress as the "Christian man" our community knows him as and would have warned him of what the Word says about living a sinful lifestyle. And anybody making the questioning points you've posted here would have been like an enemy to our family (victims of the hurt from adultery). And what do you think such questioning looks like to God. We ALL have to correct somebody at some level and not turn a sluggish eye at the opportunity.
For years, I worked in a field that required meticulous attention to the concerns of our customers or else errors could result in financial disaster for them as consumers. So, if my co-workers and I chose not to correct each other (with help, resources, pointing to references during one-on-one recap of our training) but instead turn a sluggish apathetic eye, where would that lead for our customers?? You as a human being know the correct answer to that.
And so what? As believers, we're less effective in making a difference to our surroundings than co-workers in the workplace?? Many CO-WORKERS have enough sense to not think that correcting a fellow worker doesn't mean "bossing them" or having authority. It means crisis prevention that would go both ways for each person with the others there to assist.
So, to answer your question, "On whose command? Did God himself ordain her to enact such an inquisition against her neighbors? On whose authority are you making this determination? Is Jesus no longer judge of who is true and who is false?" You've missed the point by a quantum leap. Ever heard statements like these before? Ever? "Beaver, Dad said we can't do this or that...Marsha, you know what mom told us to do before dinner. We have to follow the rules...Jim, Dad will be upset if you take the car tonight..." Hello??? Where is the inquisition against a brother/sister/neighbor? Where is the authority other than pointing out to the other person in sincere concern? You already know the answer.